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![1.1 Dimensions of physical quantity 5
M = 1000, the unit of length is decreased by a factor of L = 100, and
the unit of time is decreased by a factor of T = 1. With this change, the
units of frequency will decrease by a factor of T −1
= 1 and the units
of acceleration by a factor of LT −2
= 100. Therefore, the numerical
value of l in the system of units of measurement will be l100 instead of
1 and that of g will be 980. However, the numerical value of Π is still
remain invariant under a change of units and hence it is a dimensionless
quantity.
1.1.1 Power-law monomial nature of dimensions
What is power monomial? A function a = φ(a, b, c, .., z) is called power-
law monomial if the different parameters a, b, ..., z combined together
into one term in the form of aα
bβ
...zγ
where the exact numerical value
of the exponents α, β, ...γ are fixed by the exact nature of the quantities
a, b, ..., z as well as the dimension of a itself. For instance in the case
of a = V the volume, its dimension function φ = φ(L, M, T) so that
[a] = Lα
Mβ
T γ
and hence the exponent α = 3 and the rest of the
exponents are zero. Indeed, there is no such instances where dimension
function can be expressed as Mexp[L] or sin M log T or L2
+ M−3
etc.
We see that all of the dimensions in the Table are power-law mono-
mials, of the form
[φ] = CLa
Mb
T c
...Iz
, (1.4)
where C is a constant and the exponents a, b, .., z assumes numbers en-
suring dimensional homogeneity. In fact, this is a general result which
can be rigorously proved. Indeed, the general proof of the theorem that
the dimension function of physical quantity always obey power mono-
mial law is highly instructive and it is elegant at the same time.
Proof: To prove this we consider a class of systems described by P,
Q, and R. By virtue of equivalence of systems within a given class, the
dimensions of any quantity a depend only on the quantities P, Q, and
R, i.e.,
[a] = f(P, Q, R). (1.5)
We now find it convenient to choose three systems of units denoted by
(0), (1), and (2). To be more precise, the systems of units (1) is obtain
from (0) by decreasing P, Q and R by a factor P1, Q1 and by a factor R1
respectively. On the other hand, the system of units (2) is similarly ob-
tained from (0) simply by decreasing the units of measurement P, Q, R](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-13-320.jpg)
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![6 Scaling, scale-invariance and self-similarity
by a factor P2, Q2 and R2 respectively. Thus both the units of systems
(1) and (2) belong to the same class as the system (0) since all these three
systems differ from one another in the numerical value of their respec-
tive fundamental units of measurement. We can denote the dimension
of the physical quantity a in the systems (0) by [a] = f(P, Q, R) and
say that the numerical value of the quantity a measured in the system
(0) is equal to a. Now according to the definition of the dimensional
function, upon transition from the system (0) to the system (1) the
numerical value of the same quantity a increases by f(P1, Q1, R1) times
and hence the numerical value of the quantity a in the system (1) should
be equal to a1 = af(P1, Q1, R1). Similarly, upon transition from the (0)
to the system (2) the numerical of the quantity a is increased by a factor
f(P2, Q2, R2) and hence the numerical value of a when measured in the
system of unit of measurement (2) is a2 = af(P2, Q2, R2). Therefore,
applying the principle of equivalence we can write
a2
a1
=
f(P2, Q2, R2)
f(P1, Q1, R1)
, (1.6)
or
a2 = a1
f(P2, Q2, R2)
f(P1, Q1, R1)
, (1.7)
(0)
P, Q, R
1 1 1
P , Q , R
2 2 2
P , Q , R
2 1 2 1
Q /Q , R /R
(1)
(2)
/Q1
P
2
Figure 1.1 Schematic illustration of the transition among (0), (1) and
(2).
Hence it follows that the numerical values of the quantity a in the
systems (2) and (1) differ by f(P1, Q1, R1)/f(P2, Q2, R2). Now, let us](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-14-320.jpg)
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![8 Scaling, scale-invariance and self-similarity
P = 1 to give C = ln f(1, Q, R) or C = ln f1(Q, R). f1 now is indepen-
dent of P. Substituting this into Eq. (1.11) we obtain
f(P, Q, R) = Pα
f1(Q, R). (1.12)
Putting it back in to Eq. (1.9) gives an equation for the function f1 of
the same form as we had for f except the that the new function f1 has
one argument fewer arguments than for f. That is,
f1(Q2, R2)
f1(Q1, R1)
= f1(Q2/Q1, R2/R1). (1.13)
Repeating the the same mathematical treatment as we did for f we get
1
f1(Q, R)
∂f1(Q, R)
∂Q
=
1
Q
∂f(1, 1)
∂Q
≡
β
Q
, (1.14)
Therefore,
f1 = Qβ
f2(R). (1.15)
Upon repeating once again we finally find
f = [a] = Pα
Qβ
Rγ
, (1.16)
The constant on the right hand side is equal to one since for P = Q =
R = 1 the system of units of measurement is unchanged and there is also
no change in the numerical value of the quantity a. Thus, the dimen-
sion of a physical quantity are always expressed by a power monomial.
The power monomial nature of the dimensional quantity is often called
dimensional homogeneity and it is really the key to dimensional analy-
sis. One can hardly exaggerate the importance of this law. For instance,
owing to this law one find that the laws of physics invariant under the
choice of units of measurement.
1.1.2 Significance of power-law monomial nature of
dimensional function
What are physical laws? They are usually relations between measured
or governed quantity and a set of governing parameters. For instance,
the Newton’s second law states that the acceleration a (the governed
parameter) experienced by an object depends on two governing param-
eters such as the mass m of the object and the net force F exerted on
it and hence can be written in the form
a = a(m, F). (1.17)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-16-320.jpg)
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![1.1 Dimensions of physical quantity 9
That is, the physical laws can always be written in the form
Expression 1 = Expression 2. (1.18)
Then the power-law monomial nature of the dimensional function of
physical quantity implies that the governing parameters on the right
hand side must combine into one single term so that both sides of the
equation have the same dimensions and hence known as dimensional
analysis. Such identity could hold owing to the power-law monomial and
its homogeneity character of the dimensional functions. The following
examples might help us appreciate the importance and the significance
of the dimensional analysis.
Wingspan of an aircraft
We use the dimensional analysis to obtain the form of the dependence
of the lift for per unit wingspan on L, v and ρ where L is the width of
the wing, v is the velocity with which it is moving and ρ is air of density
through which the aircraft is moving.
Let us call the lift per unit wingspan Φ and write the following relation
Φ = kLα
vβ
ργ
, (1.19)
where k, α, β and γ are dimensionless constants. Using the dimension of
force MLT −2
we can immediately write the dimensional relation [w] =
MT −2
. Thus
MT −2
= Lα
Lβ
T −β
Mγ
L−3γ
. (1.20)
Applying the principle of equivalence we find γ = 1, β = 2 and solving
equation α + β − 3γ = 0 gives α = 1. We therefore can write
Φ = kLv2
ρ. (1.21)
This analysis suggests that we can define a dimensionless lift coefficient
η as
η =
F
1
2 Lρv2w
, (1.22)
where w is the width of the wingspan. Here the factor 1
2 is introduced
as matter of convention. Most aircraft wings have lift coefficient η in
the range 0.2 < η ≤ 0.6, depending on the design of the wing and its
orientation with respect to the motion of the aircraft.](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-17-320.jpg)
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![10 Scaling, scale-invariance and self-similarity
Speed of traveling wave through a string:
We know that the wave velocity for a medium depends on the elastic
properties of the medium and on its inertia. Let us consider the simplest
one dimension case: a stretched string. Say, of the two remote ends, one
is attached to a far distance against a rigid wall and the other end is
at hand so that we can vary the tension on the string. It is clear that
the elastic character of the string is measured by the tension F and
the inertia character is measured by the mass density µ (mass per unit
length) of the string. If we flick our wrist once we can create a pulse that
will certainly propagate with some velocity. Detailed observation with
varying tension and changing string of different µ will convince that the
speed v of the pulse depends on both F and µ. Thus the dimension
velocity can be written in the following functional form
[v] = f(T, µ). (1.23)
Using the theorem that the dimension of physical quantity must follow
power monomial character we can immediately write
[v] = Fa
µb
. (1.24)
But, the dimension of velocity is LT −1
, of F is MLT −2
and of µ is
ML−1. Plugging these into the above equation gives
a + b = 0, and a − b = 1. (1.25)
Solving it we immediately find that a = 1/2 and b = −1/2 i.e.
v = C
s
F
µ
, (1.26)
where the constant C cannot be determined by dimensional analysis
alone. Nevertheless, one can use either theory or real life data to find
out that in this case it is C = 1.
On the other hand speed of traveling wave through a solid materiel
can also be obtained using the same technique. In this case the elastic
properties of the solid is determined by the Young’s modulus Y and its
inertia is determined by the density (mass per unit volume of the solid)
ρ. Using the technique we just used we find
v =
s
Y
ρ
. (1.27)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-18-320.jpg)
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![1.2 Buckingham π-theorem 11
Oscillation of a Star:
A star undergoes some mode of oscillation. How does the frequency ω
of oscillation depend upon the properties of the star? The first step is
the identification of the physically relevant governing variables on which
the success of the dimensional analysis depend on. Certainly, the density
ρ and the radius R are important; In addition to this the gravitational
constant G may play a role also the mass m but if we assume density is
constant then we have m = ρ(4πR3
/3) and hence the mass is a redun-
dant parameter. Thus, ω is the governed parameter with dimension T −1
,
and ρ, R and G are the governing parameters with dimensions ML−3
,
L and M−1
L3
T −2
respectively. Thus applying the fact that dimension
of all physical quantities follow power-law monomial we get
T −1
= [ρ]a
[R]b
[G]c
(1.28)
= Ma−c
L−3a + b + 3cT −2c
.
Equating the exponents on both sides gives three algebraic equations
and solving them immediately gives a = c = 1/2 and b = 0 so that
ω = C
p
Gρ, (1.29)
with C is a constant. We thus see that the frequency of oscillation is
proportional to the square root of the density and independent of the
radius R. Once again the determination of C requires real data or the-
ory. But either case a dimensional analysis helps focusing on the right
parameters with to right amount of importance.
1.2 Buckingham π-theorem
The name Pi comes from the mathematical notation Π are defined here,
for historical reason, to describe dimensionless variables obtained from
the power products of governing parameters denoted by Π1, Π2, Π3...
etc. If a physical process involves investigation of a certain dimensional
physical quantity (governed quantity) that depend on n number of other
dimensional variables then the Buckingham Pi theorem guide us in a sys-
tematic way to reduce a function of n variable problems into a function of
k dimensionless variable problems if each of the k dimensional variables
of the original n variables can be expressed in terms of the n − k dimen-
sionally independent variables. Now the question is: What is dimension-
ally independent variable? A set of dimensional variables, a1, a2, ..., ak of
the total n variables, is said to have independent dimensions if none of](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-19-320.jpg)
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![12 Scaling, scale-invariance and self-similarity
the k variables have dimensions which can be represented as a product
of powers of the dimensions of the remaining variables. For instance,
density [ρ] = ML−3
, velocity [v] = LT −1
and force [F] = MLT −2
have
independent dimensions in the sense that none of these quantities can be
expressed in terms of the rest. In other words, there do not exist product
of powers of these quantities which is dimensionless. On the other hand
consider the case where force is replaced by pressure [P] = ML−1
T −2
.
In this case the dimension of pressure can be expressed in terms of the
dimension of that of velocity and density since [P] = [v2
][ρ] and hence
velocity and density have independent dimension but note the pressure.
Proof: The first step in modeling any physical phenomena are the
identification of the relevant variables, and then relating these variables
via known physical laws. For sufficiently simple phenomena, we can usu-
ally construct a quantitative relationship among the physical variables
from the first principles; however, for many complex phenomena such
an ab initio theory is often difficult, if not impossible. In these situ-
ations constructing a model in a systematic manners with minimum
input parameters that can help analyzing experimental results has been
a useful method, and one of the most powerful modeling method is
dimensional analysis. Most undergraduate have already encountered di-
mensional analysis during their course studies but without knowing its
full potential. Here we will use the dimensional analysis to capture tech-
niques for analyzing experimental or empirical data or to solve partial
differential equation which also help infer some insights about the solu-
tion. As we have already stated that the relationship found in physical
theories or experiments can always be represented in the form
a = f(a1, a2, ..., an), (1.30)
where the quantities a1, a2, ..., an are called the governing parameters.
Any investigation ultimately comes down to determining one or several
dependencies of the form Eq. (1.30). It is always possible to classify the
governing parameters a1, ..., an into two groups using the definition of de-
pendent and independent variables. Let the arguments ak+1, ..., an have
independent dimensions and the dimensions of the arguments a1, a2, ..., ak
can be expressed in terms of the dimensions of the governing indepen-
dent parameters ak+1, ..., an in the following way:
[a1] = [ak+1]α1
...[an]γ1
, (1.31)
[a2] = [ak+1]α2
...[an]γ2
,
... ... ...](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-20-320.jpg)
![1.2 Buckingham π-theorem 13
... ... ...
[ak] = [ak+1]αk
...[an]γk
.
The dimension of the governed parameter a must also be expressible in
terms of the dimensionally independent governing parameters a1, ..., ak
since a does not have independent dimension and hence we can write
[a] = [ak+1]α
...[an]γ
. (1.32)
Thus, there exist numbers α, β, and γ, such that Eq. (1.32) holds. We
can therefore a set of dimensionless governing parameters
Π1 =
a1
[ak+1]α1 ...[an]γ1
, (1.33)
Π2 =
a2
aα2
k+1...aγ2
n
,
... ... ...
Πk =
ak
[ak+1]αk ...[an]γk
,
and a dimensionless governed parameter
Π =
f(Π1([ak+1]α1
...[an]γ1
), ..., Πk([ak+1]αk
...[an]γk
, ak+1, ..., an)
[ak+1]α...[an]γ
.
(1.34)
The right hand side of this equation clearly reveals that the dimension-
less quantity Π is function of ak+1, ..., an, Π1, ..., Πk i.e.,
Π ≡ F(ak+1, ..., an, Π1, ..., Πk). (1.35)
The quantities Π, Π1, ..., Πk are obviously dimensionless, and hence
upon transition from one system of units to another inside the given
class their numerical values must remain unchanged. At the same time,
according to the above, one can pass to a system of units of measurement
such that any of the parameters of ak+1, ..., an, say for example, ak+1, is
changed by an arbitrary factor, and the remaining ones are unchanged.
Upon such a transition the first argument of F is changed arbitrarily,
and all the other arguments of the function remain unchanged as well as
its value Π. Hence, it follows δF
δak+1
= 0 and entirely analogously δF
δak+2
=
0, ... δF
δan
= 0. Therefore, the relation Eq. (1.35) is in fact represented by
a function of k arguments and proves it is independent of ak+1, ..., an
i.e.,
Π = Φ(Π1, ..., Πk), (1.36)
and the function f can be written in following special form
f(a1, ....., ak, ......., an) = aα
k+1.......aγ
nΦ((Π1, ...., Πk). (1.37)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-21-320.jpg)
![1.2 Buckingham π-theorem 15
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10
S
b
Figure 1.3 This figure shows how area S for one of the triangle of the
above figure (say the largest triangle) varies as the base b is changed.
to measure the area S of the triangle. The area S can be written in the
following form
S = S(a, b, c). (1.38)
However, the dimension of two governing parameters a and b can be
expressed in terms of c alone since we have
[a] ∼ [c] and [b] ∼ [c], (1.39)
and so is true for the governed parameter S as we can write the di-
mensional relation [S] ∼ [c2
]. We therefore can define two dimensionless
governing parameters
Π1 = sin θ = a/c and Π2 = cos θ = b/c, (1.40)
and the dimensionless governed parameter
Π =
S
c2
= c−2
S(cΠ1, cΠ2, c) ≡= F(c, Π1, Π2). (1.41)
Now it is possible to pass from one unit of measurement to another
system of unit of measurement within the same class and upon such
transition the arguments Π1 and Π2 of the function F as well as the
function itself remain unchanged. It implies that the function F is inde-](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-23-320.jpg)
![18 Scaling, scale-invariance and self-similarity
Figure 1.5 A schematic diagram showing the experiments of Bose,
Rauert and Bose. The time τ requered to fill a vessel of volume Q
and the pressure drop between the ends of the pipe P were measured
for the steady flow of various fluids through the pipe.
Bose-Rauert-Bose experiment
In 1909-1911 the physical chemists E. Bose, D. Rauert, and M. Bose
performed a series of experimental investigations based on the scheme
as described by the Fig. . The experiment consist in measuring the nec-
essary time τ to fill a container of given volume V and the pressure
drop P at the ends of a pipe for steady flow of various fluids e.g. water,
chloroform, mercury, ethyl alcohol e.t.c. through the pipe. The experi-
mental results are represented in the form of a series of curves for the
presurre drop as a function of the filling time τ for various fluids, like
those shown in Fig... The work was subsequently noted by von Karman
who performed a dimensional analysis to give a better interpretation of
the experimental results.
Von Karman’s arguments can be presented as follows. The pressure
drop P at the ends of the tubes must depends on the parameters such as
the filling time τ, the volume V of the container, the viscosity coefficient
µ and the density ρ of the fluid used. So these are the four governing
parameters which are expected to govern the phenomena in question
and hence n = 4. The dimensions of the parameters in the MLT class
can be expressed by the following relations:
[P] =
M
LT 2
, [τ] = T, [V ] = L3
, [µ] =
M
LT
, [ρ] =
M
L3
. (1.49)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-26-320.jpg)
![20 Scaling, scale-invariance and self-similarity
It is easy to see that the first three governing parameters τ, µ and V have
independent dimensions, while the dimensions of the fourth governing
parameter, ρ, are expressed in terms of the first three: [ρ] = [µ][τ][V ]2/3
.
Thus we have k = 3, so that n − k = 1, and dimensional analysis gives
Π = Φ(Π1), Π =
P
µτ−1
, Π1 =
ρ
µτV −2/3
. (1.50)
Consequently, according to the above equation, had we plot in the coor-
dinates Π and Π1, all the experimental points should fall on to a single
curve as is confirmed by the Fig.
1.3 Similarity
In most cases, before a large and expensive structure or object such as
a ship or aeroplane is built, efforts are always devoted to the testing
on model system instead of working directly with the real system. The
results of testing on model system are then used to infer the various
physical characteristics of the real object or structure under the future
working conditions. In order to successful modeling, it is necessary that
we know how to relate the results of testing on models to the actual
manufactured product. Needless to mention that if such connections are
not known or cannot be made then modeling is simply a useless pursuit.
For the purpose of rational modeling, the concept and the importance
of the similarity phenomena can hardly be over emphasized. Indeed, the
physically similar phenomena is the key to successful modeling.
The concept of physical similarity is a natural generalization of the
concept of similarity in geometry. For instance, two triangles are similar
if they differ only in the numerical values of the dimensional param-
eters, i.e. the lengths of the sides, while the dimensionless parameters,
the angles at the vertices are identical for the two triangles. Analogously,
physical phenomena are called similar if they differ only in their numer-
ical values of the dimensional governing parameters; the values of the
corresponding dimensionless parameters Π1, ..., Πm are being identical.
In connection with this definition of similar phenomena, the dimension-
less quantities are called similarity parameters.
Let us consider two similar phenomena, one of which will be called
the prototype and the other the model; it should be understood that
this terminology is just a convention. For both phenomena there is some](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-28-320.jpg)
![22 Scaling, scale-invariance and self-similarity
necessary to choose the governing parameters b
(m)
1 , b
(m)
2 , of the model so
as to guarantee the similarity of the model to the prototype:
b
(m)
1 = b
(p)
1
a
(m)
1
a
(p)
1
α1
a
(m)
2
a
(p)
2
β1
a
(m)
3
a
(p)
3
γ1
, (1.59)
and
b
(m)
2 = b
(p)
1
a
(m)
1
a
(p)
1
α2
a
(m)
2
a
(p)
2
β2
a
(m)
3
a
(p)
3
γ2
, (1.60)
whereas the model parameters a
(m)
1 , a
(m)
2 , a
(m)
3 can be chosen arbitrar-
ily. The simple definitions and statements presented above describe the
entire content of the theory of similarity: we emphasize that there is
nothing more to this. We will now give a few examples to help us cap-
ture the idea.
1.3.1 Illustrative Examples of similarity phenomena
(i) The steady motion of a body in a large vessel filled with
fluid:
To do this we consider the following assumptions:
(a) The velocity of the body is assumed to be small relative to the
velocity of sound, so that the fluid can be treated as incompressible and
hence of constant density.
(b) The direction of the velocity vector assumed to be identical for both
model and proto-type which is the case if the model and the proto-type
are geometrically similar.
In this problem, the dimensional governing parameters of the motion
are the characteristic length scale of the body such as the maximum
cross-sectional diameter D, the magnitude v of the body’s velocity, the
density ρ of the fluid, and finally its viscosity µ. The dimensions of the
governing parameters are
[D] = L, [v] = LT −1
, [ρ] = ML−3
, [µ] = ML−1
T −1
. (1.61)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-30-320.jpg)
![24 Scaling, scale-invariance and self-similarity
motion is that due to the surface waves created by the ship. The govern-
ing parameters in this case at hand will be as follows: a characteristic
length for the ship, l, the gravitational acceleration g, the density of the
fluid ρ, and the speed of the ship v. the parameter g is essential since the
gravitational force is one of the factors that controls the waves created
by the ship. The governing parameters have the following dimensions in
the LMT class:
[l] = L, [v] = LT −1
, [ρ] = ML−3
, [g] = LT −2
, (1.64)
so that n = 4, k = 3, and m = 1 which implies that there is only one
dimensionless governing parameter
Π1 =
v
l1/2g1/2
. (1.65)
This parameter is called the Froude number which is named after famous
English engineer and ship builder Wm. Froude.
Furthermore, the dimension of the drag force F in the same class
LMT is [F] = [ρ][g][l]3
so that dimensionless governed drag force is
Π =
F
l1/2g1/2
. (1.66)
Thus, since the parameter g can hardly be varied, the law for scaling
the drag forces from the model up to the prototype in the same fluid is
of the form
F(p)
= F(m)
l(p)
l(m)
3
, (1.67)
so that the drag force is proportional to the cube of the modeling scale.
Relation for the dimensionless governing parameter indicates that in
order to ensure dynamical similarity, the ratio of the model velocity to
the prototype velocity must be proportional to the square root of the
modeling scale:
v(m)
= v(p)
l(m)
l(p)
1/2
. (1.68)
If one does not neglect the role of the viscosity, a second dynamical
parameter appears - the Reynolds number Re, which is equal to ρvl/µ.
Modeling in which both similarity parameters - the Froude number and
Reynolds number - are taken into account turns out to be impossible in
the same fluid. indeed, to do this , the products vl (see example (i) above)
and v2
/l would have to be identical for both model and the prototype;
this is only possible when modeling to full scale, which makes no sense.](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-32-320.jpg)
![28 Scaling, scale-invariance and self-similarity
1.4.1 Dynamic scaling and its link to π-theorem
A function f(x, t) is said to obey dynamic scaling if one of the variable
t strictly denotes time (time developing phenomena) and if it satisfies
f(x, t) ∼ tτ
φ(x/tz
), (1.69)
where exponents θ and z are fixed by the dimensional relations [tθ
] = [f]
and [tz
] = [x] respectively, while φ(ξ) is known as the scaling function.
Sometimes it is also written in the following form
f(x, t) ∼ xω
φ(x/tz
). (1.70)
Both the form came into being as hypothesis since its deep connection
to Buckingham π-theorem has never been appreciated. Indeed, Bucking-
ham π-theorem can provide a systematic processing procedure to obtain
the dynamic scaling form and at the same time appreciate the fact that
the second form is not mathematically sound. An interesting aspect of
the structure of the dynamic scaling form given by Eq. (1.69) is that the
distribution function f(x, t) at various moments of time can be obtained
from one another by a similarity transformation
x −→ λz
x, t −→ λt, f −→ λθ
f, (1.71)
revealing the self-similar nature of the function f(x, t).
To derive it one has to know first that one of the two governing pa-
rameters can be assumed to be independent. Let us assume that t is
chosen to be an independent parameter and hence x can be expressed
in terms of t
x ∼ tz
. (1.72)
It implies that we can choose tz
as unit of measurement or yard-stick
and quantify x in terms of dimensionless quantity ξ = x/tz
. Here, the
quantity ξ is a number that tells how many tz
we need to measure x.
If t is independent quantity then we can also express f in units of tτ
to obtain yet another dimensionless quantity φ = f(x, t)/tτ
where the
exponent θ is fixed by the dimensional requirement [f] = [tθ
]. Since
φ is a dimensionless quantity its numerical value can only depend on
dimensionless quantity ξ not on depend on a dimensional quantity t. This
is reminiscent of that fact that the numerical value of S/c2
is independent
of the size of the hypotenuse c for fixed value of the lesser angle θ. We
can then immediately obtain the scaling form given by Eq. (1.69). On
the other hand, had we choose x to be independent parameter instead of](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-36-320.jpg)
![1.4 Self-similarity 29
t then following the same argument we would have the following scaling
form
f(x, t) ∼ xω
φ(t/xν
), (1.73)
where ν = 1/z. Note that if we express f in unit of x then t should also
be expressed in unit of x.
There are many phenomena which physicist often investigate are not
static rather evolve probabilistically with time. In such cases too one
often is interested to see if certain distribution function of the systems at
different times are similar. In fact, we use the very definition of similarity
but not two different systems but of the same system at various moments
of time. Emergence of dynamic scaling is in fact deeply rooted to the
Buckingham’s π-theorem. Note that one can derive the dynamic scaling
form if we find that one of the two variables of the function f(x, t), say
t, has independent dimension and the dimension of the other variable
x can be expressed in terms of t alone i.e. [x] ∼ [tz
]. This means that
the function f can also be expressed in terms of the dimension of t and
satisfy the dimensional relation [f] = [tθ
]. Following the steps of the
Buckingham’s π-theorem can immediately led to the dynamic scaling
form. Existence of dynamic scaling also means existence temporal self-
similarity since it implies that the dimensionless quantities at different
time must be identical. Indeed, one way of verifying the dynamic scaling
is to plot dimensionless variables f/tθ
as a function of x/tz
of the data
extracted at various different time. Then if all the plots collapse onto a
single universal curve then it is said that the systems at different time
are similar and it obeys dynamic scaling. Essentially such systems can
be termed as temporal self-similarity since the the same system is similar
at different times.
1.4.2 Finite-size scaling and its link to π-theorem
There exists yet another scaling hypothesis, known as the finite-size scal-
ing (FSS), that has been extensively used as a very powerful tool for es-
timating finite size effects specially in the second order phase transition
near the critical temperature T . The various response functions, typi-
cally the second derivative of the free-energy F, in second order phase
transition diverges. Such transitions are classified by a set of critical ex-
ponents which characterize the critical point. The best known example
of second order phase transition is the paramagnetic to ferromagnetic](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-37-320.jpg)
![1.5 Scale-invariance: Homogeneous function 33
where prime indicates derivative with respect to the argument of the
corresponding function. We now set µ = 1 and g′
(1) = p then we can
immediately write
1
g(λ)
g′
(λ) =
p
λ
. (1.91)
Integrating it we find
ln[g(λ)] = p ln[λ] + c, (1.92)
or
g(λ) = ec
λp
. (1.93)
Now from the above equation we have g′
(λ) = pec
λp−1
and the definition
g′
(1) = p implies that the integration constant c has the value zero. Thus
g(λ) ∼ λp
, (1.94)
and it makes the proof complete.
The power-law type distributions f(x) ∼ x−α
are regarded as scale-
free because f(λx)/f(x) depends only on λ not on x and hence their
distribution requires no characteristic or typical scale. The idea is that
if the unit of measurement of x is increased (decreased) by a factor of λ
then the numerical value of f is decreased (increased) by a factor of g(λ)
but the overall shape of the function remains invariant. It is called scale-
free also because it ensures self-similarity in the sense that the function
looks the same whatever scale we look at it. One immediate consequence
of the scale-invariant function f(r) is that if we know its value at one
point say at r = r0, and we know the functional form of g(λ), then the
function f(r) is known everywhere. This follows because any value of r
can always be written in the form r = λr0, and
f(λr0) = g(λ)f(r0). (1.95)
The above equation says that the value f(r) at any point is related to
the value of f(r0) at a reference point r = r0 by a simple change of scale.
Of course, this change of scale is, in general, not linear.
1.5.1 Scale invariance: Generalized Homogeneous
functions
We shall see latter in the static scaling hypothesis where we shall use the
fact that the thermodynamic potentials are homogeneous of the form A
function f(x, y) of two independent variables x and y is said to be a](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-41-320.jpg)
![2.1 Stochastic Process 43
2.1 Stochastic Process
Stochastic process provides the theoretical framework for studying non-
equilibrium statistical mechanics. To be able to understand and appreci-
ate the meaning of a stochastic process it is essential to understand the
concept of random variable as it lies at the very heart of its definition.
Stochastic or random variable x is defined by (i) a set of possible values
x1, x2, .., xn which we may call range or set of states and (ii) probabil-
ity distribution over this set. Consider a simple example of a dice to
illustrate the two points mentioned above. In each throw, the number
in the upper face corresponds to the variable x, with possible outcomes:
{1, 2, 3, 4, 5, 6} and the probabilities attached with each of the outcomes
which is 1/6 (for an honest dice) in the present example. It is worth-
while to mention that the range or the set of states may be discrete or
continuous, finite or infinite. If the set is discrete (as the case of dice,
or coin) then the probability distribution will be given by a set of non
negative numbers such that the attached probabilities satisfies
X
n
Pn = 1. (2.1)
On the other hand, if the range is continuous within an interval [a, b] over
the x axis then the probability distribution is given by a non negative
function P(x) 0 which is normalized in the sense that
Z b
a
P(x)dx = 1, (2.2)
where the integral extends over the whole range. The probability that x
has a value between x and x + dx is P(x)dx.
Once a stochastic variable x has been defined, an infinity of other
stochastic variables can be derived from it. For instance, all quanti-
ties y = f(x) obtained by some mapping f is again a stochastic vari-
able which always indicates some conceptual method. These quantities
y could also be functions of an additional variable t, i.e.
y(t) = f(x, t), (2.3)
where t is the time and typically it represents the realization of the pro-
cess. The function y(t) describes stochastic process if t stands for time.
Thus a stochastic function is simply a function of two variables, one of
which is the time t and the other is a stochastic variable x as defined
above. In other words, systems which evolve probabilistically in time or
systems in which there exists a certain time-dependent random variable](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-51-320.jpg)
![44 Stochastic process, random walk and diffusion
x(t) then it is regarded as stochastic process. We can measure the values
of x to find x1, x2, ..., xn etc at times t1, t2, ..., and construct a joint prob-
ability P(xn, tn; xn−1, tn−1; , ..., x1, t1). The simplest kind of stochastic
process is that of complete independence where joint probability satisfies
P(xn, tn; xn−1, tn−1; , ..., x1, t1) = Πip(xi, ti), (2.4)
which mean that that value of xi at time ti is completely independent
of its value in the past or future.
Let y denote a variable in which we are interested in. It could be, for
instance, the position or the velocity of a particle. If y were a determin-
istic quantity obeying a dynamical law, we could construct a function of
time y = y(t), which determines a well defined value of y at each time t.
Given appropriate initial data y(0) = y0 we get what is called a trajec-
tory, or an orbit. But if y is a random variable such a function does not
exist. At each successive time, the variable y can take any value within
the sample space with some probability attached to each member of the
sample space. Another way of looking at this situation is as follows. In
a first experiment, the system starting at y = y0 follows a well defined
orbit or sample path: y1(t) as described above. But if the experiment is
repeated, starting from the same initial condition, the sample path of
the random variable (in contrast to deterministic one) will be different in
each realization. However, at every time t, there is a certain probability
[0, 1] attached to every possible outcome y of the sample space. If we
assume y is a continuous variable, we should rather speak about prob-
ability densities. To be more precise, we define the probability density
p(y, t) of the random variable y at time t by stating that the probability
of finding the value of the variable in the interval (y, y + dy) is p(y, t)dy.
Note that the concept of a random variable includes the deterministic
variable as a special case.
So the question is: How do we recognize if a process is stochastic in
nature or not? As we already mentioned that a system is stochastic if it
evolve probabilistically in time and if we can construct we assume that
a set of joint probability densities
Pn(xn, tn; xn−1, tn−1; xn−2, tn−2; ..., x1, t1), (2.5)
exists which describe the system completely. In terms of these joint prob-
ability density functions, one can also define the conditional probability
densities
Pk|m(ym+k, tm+k; ...; ym+1, tm+1|ym, tm; ...; y1, t1), (2.6)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-52-320.jpg)
![2.1 Stochastic Process 45
as the joint conditional probability density that the stochastic variable
y has values (ym+k, tm+k; ...; ym+l, tm+l) given that (y1, t1; ...; ym, tm).
These definitions are valid independently of the ordering of the times,
although it is usual to consider only times which increases from right to
left i.e.,
tn ≥ tn−1 ≥≥ tn−2 ≥ ... ≥ t1. (2.7)
The concept of an evolution equation leads us to consider the conditional
probabilities as predictions of the future values of x(t), given the knowl-
edge of the past. To define the concept of a general stochastic process we
need to know at least all possible joint probabilities. If such knowledge
does define the process then it is known as a separable stochastic pro-
cess. All the process that will be considered in this book will be assumed
to be of this kind.
Let x be a stochastic variable defined in the range (−∞, ∞) and with
the distribution P(x). We define the average of some function f(x) over
the distribution as
f(x) =
Z ∞
−∞
f(x)P(x)dx. (2.8)
2.1.1 Essential Nomenclatures and Definitions:
Before introducing various quantities of interest we find it worthwhile to
discuss very briefly about the fundamental concept of joint probability
and conditional probabilities which is again central to the rest of our
discussion.
• Joint Probabilities
We all know that the occurrence of mutually exclusive events are re-
lated to the concept of nonintersecting sets. An event ω which satisfies
ω ∈ A will only satisfy ω ∈ A ∩ B if ω ∈ B as well. Thus,
P(A ∩ B) = P{(ω ∈ A) and (ω ∈ B)}
and P ∩ B) is called the joint probability that the event ω is contained
in both classes, or, alternatively, that both the events ω ∈ A and ω ∈ B
occur. Joint probabilities occur naturally in the text of this course in
two ways:
i) When the event is specified by a vector, e.g., m mice and n cats. The
probability of this event is the joint probability of [m mice (and of
any number of cats)] and [n cats (and of any number of tigers)].](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-53-320.jpg)
![50 Stochastic process, random walk and diffusion
We assume now that the transition probability from a state y1 to a
different state y2 in a sufficiently short time interval t2 − t1 = ∆t is
proportional to ∆t; thus
P(y2, t + ∆t|y1, t) = Aδ(y2 − y1) + ∆tω(y1 → y2; t) + O((∆t)2
), (2.20)
where ω(y1 → y2; t) is the transition probability per unit time. The
first term on the right hand side represents the probability that the
random variable remains in y1 during the interval ∆t.; the factor A
is determined by the normalization condition Eq. (2.17) in Eq. (6.19).
That is, we integrate Eq. (6.19) with respect to y2 for which the left
hand side is equal to one and the right hand side is A + ∆ta(y1) where
a(y1) =
R
dy2ω(y1 → y2; t)
A = 1 − ∆ta(y1), (2.21)
is the probability that no transition occurs during the time interval
Deltat. We now write the Chapman-Kolmogorov equation Eq. (2.85),
setting y3 = y, t3 = t + ∆t, y2 = y′
, t2 = t to obtain
P(y, t + ∆t|y1, t1) =
Z
dy′
P(y, t + ∆t|y′
, t)P(y′
, t|y1, t1). (2.22)
We then expand the right hand side to first order in ∆t using Eq. (6.19)
and find the following:
P(y, t + ∆t|y1, t1) =
Z
dy′
[Aδ(y − y′
) + ∆tω(y′
−→ y)]P(y′
, t|y1, t1)
=
Z
dy′
[1 − ∆ta(y′
)]δ(y′
− y)P(y′
, t|y1, t1) (2.23)
+ ∆t
Z
dy′
ω(y′
−→ y)]P(y′
, t|y1, t1).
Further manipulation gives:
P(y, t + ∆t|y1, t1) − P(y, t|y1, t1)
∆t
= −
Z
dy′
a(y′
)δ(y′
− y)P(y′
, t|y1, t1)
+
Z
dy′
ω(t′
−→ y)]P(y′
, t|y1, t1),
where a(y) =
R
dsω(y −→ s, t) and hence
∂P(y, t|y1, t1)
∂t
= −
Z
dsω(y → s; t)P(y, t|y1, t1)
+
Z
dsω(s → y, t)P(s, t|y1, t1). (2.24)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-58-320.jpg)
![2.3 Random walk (RW) problem 53
the expression for P(m, N) by using Stirling’s formula
n! =
√
2πn(
n
e
)n
, (2.29)
and thus
ln n! = (n + 1/2) ln n − n +
1
2
ln[2π]. (2.30)
Accordingly, in the limit N −→ ∞ and m N we have
ln P(m, N) ≈ (N + 1/2) ln N − N −
N + m + 1
2
ln(N + m)/2
(2.31)
+
N + m
2
−
N − m + 1
2
ln(N − m)/2 +
N − m
2
−
1
2
ln 2π − N ln 2. (2.32)
Simple algebraic manipulation immediately gives
ln P(m, N) ≈ (N + 1/2) ln N −
N + m + 1
2
ln N/2(1 + m/N)(2.33)
−
N − m + 1
2
ln N/2(1 − m/N) −
1
2
ln 2π − N ln 2.
Using the series expansion
ln(1 ± x) ≈ ±x − x2
/2, (2.34)
Eq. (2.33) becomes
ln P(m, N) ≈ (N +
1
2
) ln N −
N + m + 1
2
(ln
N
2
+
m
N
−
m2
2N2
)
−
N − m + 1
2
(ln
N
2
−
m
N
−
m2
2N2
) −
1
2
ln 2π − N ln 2.
(2.35)
Further simplifying the right hand side of this equation we obtain
ln P(m, N) ≈ −
1
2
ln N −
1
2
ln(2π) + ln 2 −
m2
2N
. (2.36)
In other words, for large N we have asymptotic formula
P(m, N) = 2/
√
2πN exp[−
m2
2N
]. (2.37)
Now when N is very large it is convenient to introduce a continuous
variable x = ml instead of m where l is the length of one step. Fur-
thermore, if we consider intervals ∆x along the straight line which are
large compared with the length of a step we can ask the probability](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-61-320.jpg)
![54 Stochastic process, random walk and diffusion
P(x)∆x that the particle is likely to be in the interval x, x + ∆x after
N displacements. We clearly have
P(x, N)∆x = P(m, N)(∆x/2l), (2.38)
since m can take only even or odd values depending on whether N is
even or odd. Combining all these we obtain
P(x, N) =
1
√
2πNl2
exp[−
x2
2Nl2
]. (2.39)
Suppose now that particles makes n = N/t displacement per unit time.
Incorporating this in to the above equation we can write it in the fol-
lowing Gaussian form
P(x, t)dx =
1
√
4πDt
exp[−
x2
4Dt
]dx, (2.40)
where we have written
D = nl2
/2. (2.41)
Later we shall find out that the quantity D introduced here is identical
with the diffusion coefficient of the given system. Of course, it comes
as no surprise that this probability distribution given by Eq. (2.40) is
symmetric and that the walker may returns to zero as many time as it
likes in time which is obviously different at each event. The statistical
analysis of such event too will be discussed in an attempt to quantify
the probability of return to zero as a function of time.
2.4 Master equation for RW problem
The RW problem is one of the best known example of the Markov pro-
cess as we shall show that it can be formulated as a Markov chain of
transition probabilities. We start with discrete-time random walk and
denote P(n∆, sτ) as the probability density for the walker to be at po-
sition x = n∆ at time t = sτ where we assumed that the walk is on a
one dimensional lattice of lattice constant ∆ and that the time between
steps is τ. Then according to Eq. (2.15) we can write
P(n∆, (s + 1)τ) =
∞
X
m=−∞
P(m∆, sτ)P1/1(n∆, (s + 1)τ|m∆, sτ), (2.42)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-62-320.jpg)
![2.4 Master equation for RW problem 57
1d lattice), then the average displacement would be given by the sum of
each individual displacement divided by N i.e.
x(n) =
1
N
N
X
i
xi(n) =
1
N
N
X
i
[xi(n − 1) ± ∆].
Iterating it n − 1 times we are left with only plus/minus terms which
becomes zero and it states that on the average particle stays where it is.
It just shows the fact that the mean displacement of the walker is zero.
To find out how particles spread out with time one has to look into the
mean-square displacement instead of the mean displacement. First we
square the displacement
x2
(n) = x2
(n − 1) ± 2∆x(n − 1) + ∆2
.
Again the plus/minus terms cancel and hence the mean square displace-
ment is given by
x2
(n) = x2
(n − 1) +∆2
.
Assuming that xi(0) = 0 we find xi(1) = ∆2
, xi(2) = 2∆2
...,
xi(n) = n∆2
. Furthermore, assuming that each step occurs in time
τ we have t = nτ. Consequently, we have
x2
=
t
τ
∆2
= 2
∆2
2τ
t.
Introducing, D = ∆2
2τ we immediately find
x2
= 2Dt. (2.54)
It can also be confirmed using the diffusion equation. We can multiply
on both sides by x2
and integrate it over the whole regions space
Z ∞
−∞
dxx2 ∂P(x, t)
∂t
=
Z ∞
−∞
dxx2
D
∂2
P(x, t)
∂x2
. (2.55)
We can write the left hand side of the above equation as
∂
∂t
x2
.
Integrating by parts and using the facts that P(x, t) and its derivative
vanish at x −→ ±∞ we obtain
Z ∞
−∞
dxx2
D
∂2
P(x, t)
∂x2
= −2D
Z ∞
−∞
dxx
∂P(x, t)
∂x
= 2D
Z ∞
−∞
dxP(x, t).](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-65-320.jpg)
![58 Stochastic process, random walk and diffusion
But P(x, t) is normalized to one, the result is simply equal to 2D and
hence
∂
∂t
x2
= 2D, (2.56)
which we can integrate to obtain
x2
(t) = 2Dt, (2.57)
and hence
xrms ∼ t
1
2 , (2.58)
as expected. Here we have used the fact that at t = 0, the probability
distribution is a δ function. The proportionality of root mean square
displacement of the Brownian particle to the square root of time is a
typical consequence of the random nature of the process which is in
sharp contrast with the result of usual mechanical deterministic and
reversible motion where displacement varies linearly with time t.
2.5 Diffusion and Fick’s Law
Typically, diffusion phenomenon is associated with the movement of
atoms or molecules within a gas, a liquid, or a solid over a more or less
long distance. For instance, we may denote the density of chemically
identical particles suspended in the fluid by a concentration variable
C(x, t) defined as the number of particles per unit volume at time t.
However, we assume that the total number of particles in the system is
a conserved quantity. However, the spatial variation in the number of
particle result in concentration gradient i.e., a particle current. Now let
there be a total of N0 particles suspended in a liquid. Let us suppose
that we know the numbers of particles at time t at two points on a line,
i.e. at point x and at x + δ, are N(x, t) and N(x + ∆, t) respectively. We
then want to know what the net flux is across an imaginary boundary
separating these two points. At time t + τ half of the particles at x will
move one step to the right, and half of the particles at x+δ will move to
the left. The net number of particles crossing the boundary will simply
be
1
2
[N(x + ∆, t) − N(x, t)],
and hence the net fluxt is
Jx = −
1
2Aτ
[N(x + ∆, t) − N(x, t)], (2.59)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-66-320.jpg)
![2.5 Diffusion and Fick’s Law 59
since it is the number of particles per unit area per unit time. Let C(x, t)
be the number of particles per unit volume which is equal to N(x, t)/Aδ
since we know that there N(x, t) particles within the volume Aδ. In
an attempt to incorporate the definition of concentration C(x, t) in Eq.
(2.59) we multiply on its right hand side by ∆2
/∆2
and re-write it as
Jx = D
1
δ
[C(x + ∆) − C(x)], (2.60)
where we set D = ∆2
/2τ. In the limit ∆ → 0 we can express it as
Jx = D
∂C(x)
∂x
. (2.61)
This is known as the Ficks first law which states that the net flux at a
point is proportional to the concentration gradient.
Now, to incorporate time t we consider a thin box and check how
the concentration changes within a given time interval. We will assume
that the number of particles N0 is a conserved quantity. We must ap-
preciate the fact that the concentration change within some time period
1/τ[C(x, t + τ) − C(x, t)] will definitely be a function of the fluxes at
either side of the box. Note that the net number of particles crossing at x
and at x+δ are J(x)Aτ and J(x+∆)Aτ respectively. So, the number of
particles accumulating in the box of volume Aδ is −[J(x+∆)−J(x)]Aτ
and hence the concentration change within the time interval τ is
1
τ
[C(x, t + τ) − C(x, t)] = −
1
τ
[J(x + ∆) − J(x)]
Aτ
A∆
. (2.62)
Again allowing the limit τ → 0 and ∆ → 0 gives
∂C(x, t)
∂t
= −
∂J(x)
∂x
. (2.63)
Substituting the Fick’s first in the above expression we can immediately
obtain the Fick’s second law which in fact is the diffusion equation
∂C(x, t)
∂t
= D
∂C(x, y)
∂x2
. (2.64)
Obviously, C(x, t)/N0 correspond to P(x, t) thereby implies that the
random walk problem is intimately related with the phenomenon of dif-
fusion.
Einstein’s approach to model Brownian motion is indeed based on
”random or drunkards walk” formalism. There are two points major
points to Einstein’s approach to the problem of Brownian motion. (i) The
motion is caused by exceedingly frequent impacts on the pollen grains](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-67-320.jpg)
![62 Stochastic process, random walk and diffusion
Incorporating Eq. (2.65) in the diffusion equation we get
∂P(k, t)
∂t
= −Dk2
P(k, t). (2.67)
This equation can be solved to yield
P(k, t) = P(k, 0)e−k2
Dt
, (2.68)
where P(k, 0) =
R ∞
−∞ eikx
δ(x)dx = 1. The inverse Fourier transform
yields
P(x, t) =
1
2π
Z ∞
−∞
dke−ikx
P(k, t) (2.69)
=
1
2π
Z ∞
−∞
dke−ikx−k2
Dt
=
1
2π
Z ∞
−∞
dkeDt(k2
2ikx/(2Dt)−x2
/(4D2
t2
)+x2
/(4D2
t2
))
=
1
2π
Z ∞
−∞
dke−Dt((k+ix)(2Dt))2
+x2
/(4D2
t2
))
=
1
2π
Z ∞
−∞
dke−Dt((k+ix)(2Dt))2
−x2
/(4Dt)
=
r
1
4πDt
e−x2
/(4Dt)
,
which is identical to the solution of the master equation of the discrete
space continuous time RW problem when the diffusion coefficient is set
to 1.
2.7.2 Scaling solution
Diffusion equation for the probability density function P(x, t) suggest
that it is a function of three variable i.e.
P = P(x, t, D), (2.70)
where x and t are the explicit variables and the diffusion constant D
is an implicit one. In order to solve the diffusion equation let us first
invoke the idea of simple dimensional analysis. Within the MLT class,
their dimensions are
[x] = L [D] =
L2
T
[t] = T and [P] = L−1
. (2.71)
The above dimensional relation implies that out of those three governing
parameters only two have independent dimensions while the dimension](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-70-320.jpg)
![2.7 Solutions of diffusion equation 63
of the third governing variable can be expressed in terms of the first
two. For instance, we can take x and t as an independent variable so
that the third D can be expressed as [D] = [x2
/t]. It is, however, more
convenient to choose the dimensions of D and t as independent one and
hence we have the following dimensional relation
[x] = [
√
Dt], (2.72)
which is consistent with the relation for the root-mean square displace-
ment x2
(t) = 2Dt. Consequently, the governed parameter P can also
be expressed in terms of D and t alone as we can write the following
dimensional relation
[P(x, t)] = [Dt]θ
with θ = −1/2. (2.73)
The exponent θ = −1/2 is required by using the normalization con-
dition
R ∞
−∞
P(x, t)dx = 1. Following the Pi theorem we can define a
dimensionless governing parameter
ξ =
x
√
Dt
, (2.74)
and a dimensionless governed parameter
Π =
P(x, t)
(Dt)θ
=
1
Dt
P(ξDt, t) = φ(Dt, ξ). (2.75)
Note that numerical value of Π remain invariant even if we change the
unit of measurement of t by an arbitrary factor but remain within the
same class since Π is a dimensionless quantity and hence we can write
φ(Dt, ξ) = φ(ξ). (2.76)
Thus the dimensional analysis and the Buckingham Pi theorem reveals
that the probability density function P(x, t) assumes a simple universal
scaling form
P(x, t) ∼
1
√
Dt
φ(x/
√
Dt), (2.77)
where φ(ξ) is the dimensionless universal scaling function. The structure
of this scaling form is highly instructive as it greatly simplifies further
analysis.
We now take derivative of Eq. (2.77) with respect to t once and with
respect to x twice to obtain
∂P(x, t)
∂t
= −
1
2
√
Dt3/2
h
ξ
dφ(ξ)
dξ
+ φ(ξ)
i
, (2.78)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-71-320.jpg)
![64 Stochastic process, random walk and diffusion
and
∂2
P(x, t)
∂x2
=
1
(Dt)3/2
d2
φ(ξ)
dξ2
. (2.79)
Upon substitution of these into the diffusion equation we find that the
solution of the partial differential equation reduces to the solution of an
ordinary differential equation for the function φ(ξ) given by
h d2
dξ2
+
ξ
2
d
dξ
+
1
2
i
φ(ξ) = 0. (2.80)
Solving this equation subject to the normalization condition
R ∞
−∞ φ(ξ)dξ =
1 is now enormously simplified. We can re-write Eq. (2.80) as
d
dξ
hdφ
dξ
+
ξ
2
φ
i
= 0. (2.81)
It means
dφ
dξ
+
ξ
2
φ = C, (2.82)
where the integration constant C = 0 since dφ(0)/dξ = 0 and hence the
solution for φ is
φ = A exp[−ξ2
/4], (2.83)
where A is the integration constant fixed by the normalization condition.
Substituting this into the normalization condition for φ immediately
gives A = 1/
√
4π and therefore
P(x, t) =
1
√
4πDt
exp[−
x2
4Dt
], (2.84)
which is the same as the one we found using Fourier transform as well as
the one obtained by the probabilistic approach in the continuum limit.
It is surprising that the exact and scaling solution are identical although
the scaling solution generally means the solution in the long-time limit
where it usually assumes a simple universal form. However, advantage
of using scaling theory is that it provides much deeper insight into the
problem which the reader will soon be able to appreciate.](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-72-320.jpg)
![2.7 Solutions of diffusion equation 65
2.7.3 Scaling Properties of the RW problem
Scale-invariance
Let us look into another interesting aspect of the RW problem. Recall
the Chapman-Kolmogorov equation
P1/1(y3, t3|y1, t1) =
Z
dy2P1/1(y3, t3|y2, t2)P1/1(y2, t2|y1, t1), (2.85)
and assume that at time t1 = 0 the walker was at position y1 = 0. Let
us now choose that y3 = x; t3 = 2t and y2 = y, t2 = t and express
the transition probability in terms of the probability as a function of
the size of the jump and the corresponding size of the time interval
P1/1(y3, t3|y1, t1) = P(y3 − y1, t3 − t1). Using these we can re-write the
Chapman-Kolmogorov equation as
P(x, 2t) =
Z ∞
−∞
dyP(x − y, t)P(y, t) =
1
√
4πD2t
exp[−x2
/(4D2t)],
(2.86)
which is an interesting identity for the RW problem. This identity states
that the transition from (0, 0) to (x, 2t) is identical to the transition
from (y, t) to (x − y, t) (see Fig. for illustration). We thus find that
when the displacements are viewed with only half the time resolution the
increments of the particle position are still a Gaussian random process
with zero mean x = 0 however at the expense of increased variance
x2
= 4Dt. The argument can easily be extended to generalize the
identity to the following form
P(x, λt) =
1
√
4πDλt
exp[−x2
/(4Dλt)]. (2.87)
To illustrate the idea developed above we show the walker’s position
when observed every microscopic step figure ... and in figure .... when ob-
served every fourth microscopic step where in both the cases the walker
made in total 10, 000 independent steps with mean x = 0. The dif-
ference lies in the fact that in the later case is that each of the increments
is the sum of 4 independent steps and there is hardly any distinction in
the two figures at least in the statistical sense.
Let us now look into the mater when in addition to time the position
is also rescaled. To do this let us replace x by λa
x and t by λb
t in the
solution for P(x, t) and find that
P(λa
x, λt) = λ− 1
2
h 1
√
4πDt
e−λ2a−1 x2
4Dt
i
. (2.88)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-73-320.jpg)
![3.1 Master Equation for Binary Fragmentation Process 75
3.1 Master Equation for Binary Fragmentation
Process
To derive the stochastic theory of fragmentation, we use an approach
based on a probability balance method devised by Janossy who described
it as the regeneration point or the first collision method. It was subse-
quently extended to cover problems in neutron transport theory by Pal
and Bell and also to radiation damage and to the theory of fragmenta-
tion by M. M. R. Williams. According to the first collision method we
define the conditional probability P(N, t|x) as the probability of there
being N particles of any size at time t provided initially there was one
particle of size x. We also define a(x) as the fragmentation rate [T −1
]
for particles of size x. Furthermore, we define
χi(x −→ x1, x2, ..., xi)dx1dx2...dxi, (3.1)
as the probability that describes the fragmentation of particles of size x
leading instantaneously to i new particles with size in the ranges [x1, x1+
dx1], [x2, x2 + dx2], ..., [xi, xi + dxi]. We assume that the fragmentation
process is characterized by a fragmentation rate a such that a exp[−a(t−
t′
)]dt′
is the probability that a particle will experience a breaking action
in dt′
and undergo no further breaking action in the subsequent time
t−t′
. We now use the Janossy’s method to formulate a balance equation
for P(N, t|x) for binary fragmentation
P(N, t|x) = δN,1e−a(x)t
+ a(x)
Z t
0
dt′
e−a(x)(t−t′
)
Z
dx1
Z
dx2 (3.2)
×
N
X
N1=0
χ2(x → x1, x2)P(N1, t′
|x1)P(N − N1, t′
|x2),
where Eq. (3.2) is derived by tracing the initial particle x, breaks into
two fragments that subsequently produce two independent cascades con-
taining N1 and N−N1 fragments respectively. The first term on the right
hand side describes the probability that no action to break particle takes
place i.e. particles avoid breaking while the second term is the probabil-
ity that as a result of fragmentation two (or three in the case of ternery)
new particles are produced at each time event.
The probability balance equation Eq. (3.2) can be re-written more
concisely in terms of the generating function. We define the generating](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-83-320.jpg)
![76 Stochastic theory of fragmentation
function as
G(Z, t|x) =
∞
X
N=0
ZN
P(N, t|x), (3.3)
and incorporating it into Eq. (3.2) gives
G(Z, t|x) = Ze−a(x)t
+ a(x)
Z t
0
dt′
e−a(x)(t−t′
)
Z Z
dxdy
× χ2(x → x1, x2)G(Z, t′
|x1)G(Z, t′
|x2). (3.4)
In deriving it we have first swapped the summations and then used the
change of variable as follows
∞
X
N=0
N
X
N1=0
ZN
P(N1, t|x1)P(N − N1, t|x2) (3.5)
=
∞
X
N1=0
∞
X
N=N1
ZN1
P(N1, t|x1)P(N − N1, t|x2)
= G(Z, t|x1)G(Z, t|x2). (3.6)
We can further simplify Eq. (3.4) by differentiating it with respect to
time t to give
∂
∂t
+ a(x)
G(Z, t|x) = a(x)
Z
dx1
Z
dx2χ2(x → x1, x2; t)
× G(Z, t|x1)G(Z, t|x2). (3.7)
This is a non-linear integro-differential equation for the generating func-
tion G and an exact solution of this equation is very much unlikely
except in very special cases.
To make the problem analytically solvable, we shall deal with the
equations for the mean number of particles N(x, t) at time t given
that initially there was one particle of size x. It can be obtained from
the definition of the generating function via
∂G(Z, t|x)
∂Z
|Z=1 =
∞
X
N=0
NP(N, t|x) = N(x, t) . (3.8)
Incorporating it in Eq. (3.7) yields
h ∂
∂t
+ a(x)
i
N(x, t) = a(x)
Z Z
dx1dx2χ2(x → x1, x2)
× [ N(x1, t)+ N(x2, t) ]. (3.9)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-84-320.jpg)
![3.1 Master Equation for Binary Fragmentation Process 77
Rearranging the dummy integration variables we can re-write it as
∂
∂t
+ a(x)
N(x, t = a(x)
Z Z
dx1dx2 N(x1, t) (3.10)
× [χ2(x → x1, x2) + χ2(x → x2, x1)],
where, χ2(x → x1, x2) is the probability that a particle of size x breaks
instantaneously into 2 new particles of size x1 and x2.
Let us introduce a daughter distribution function b(x1|x) as
b(x1|x) =
Z
dx2[χ2(x → x1, x2) + χ2(x → x2, x1)], (3.11)
so that we can express Eq. (??) in a more convenient and concise form
h ∂
∂t
+ a(x)
i
N(x, t) =
Z x
0
dx1b(x1|x) N(x1, t) . (3.12)
We now find it convenient to introduce the density field defined as
N(x, t) =
Z
c(x → y, t)dy, (3.13)
so that c(x → y, t)dy is the mean number of particles in the size range
(y, y + dy) at time t, given that there was one particle of size x at time
t = 0. It is note worthy to mention here that the variable y in Eq. (3.13)
is a dummy one. Substituting the definition of the density field given by
Eq. (3.13) into Eq. (3.12) gives
Z
dy
∂c(x → y, t)
∂t
= −
Z
dya(x)c(x → y, t)
+
Z
dya(x)
Z x
0
dy1b(y1|x)c(y1 → y, t), (3.14)
and hence we can write
h ∂
∂t
+ a(x)
i
c(x → y, t) = a(x)
Z x
0
dy1b(y1|x)c(y1 → y, t). (3.15)
It is noteworthy to mention that the initial conditions in the two terms
of Eq. (3.16) are different. Hence it is not convenient to find any physical
picture of the fragmenting system since it can not keep track of the total
mass of the system which should be a conserved quantity. We therefore
attempt to find the adjoint of Eq. (3.15) with respect to the variable x.
To this end, we find it appropriate to define an operator L(x, t) so that
we can express Eq. (3.15) as
L(x, t)c(x → y, t) = 0. (3.16)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-85-320.jpg)
![3.2 Ternery fragmentation 83
It is clear that rearranging the dummy integration variables we can write
[
∂
∂t
+ a(x)] N(x, t =
Z
dx1b(x1|x) N(x1, t) , (3.49)
if we define the daughter distribution function as follows
b(x1|x) =
Z
dx2
Z
dx3
h
χ3(x → x1, x2x3)
+ χ3(x → x3, x2, x1) + χ3(x → x2, x3, x1)
i
. (3.50)
We thus see that Eq. (3.49) is exactly the same as Eq. (3.12). It means
that if we follow the same procedures as we did for binary case we will
have the same equation for density field given by Eq. 3.25). We thus
see that binary, ternery or even higher order fragmentation equations
gave the same form if we express them in terms of daughter distribu-
tion function b(x|y) and the breakup rate a(x). However, their daughter
distribution functions are different. For instance in the case of ternery
fragmentation we have
b(x|y) =
Z y−x
0
dx1
h
ω(x → x1, x, y − x − x1) (3.51)
+ ω(x → x1, y − x − x1, x) + ω(x → y − x − x1, x, x1)
i
.
Integrating it with respect to x gives
Z y
0
dxb(x|y) = 3 and
Z y
0
dxxb(x|y) = y, (3.52)
which is once again required by the conservation of mass principle. We
now introduce the symmetric kernel F(x, y, z) as
b(x|y) = 3
R y−z
0
F(x, z, y − x − z)dz
a(y)
, (3.53)
and hence upon integrating on both side with respect to x gives
a(y) =
Z y
0
Z y−z
0
F(x, z, y − x − z)dz. (3.54)
Finally, we can write ternery fragmentation equation as
∂c(x, t)
∂t
+ c(x, t)
Z x
0
dy
Z x−y
0
F(y, z, x − y − z)dz
= 3
Z ∞
x
dyc(y, t)
Z y−x
0
dzF(x, z, y − x − z). (3.55)](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-91-320.jpg)
![3.2 Ternery fragmentation 85
In the study of fragmentation and aggregation processes one of the
most important and useful quantity is the concept of the particle size
distribution function c(x, t) also known as concentration of particle of
size x at time t. It is defined as the number of particles those have the
size range within [x, x+dx] per unit mass at time t. That is it bears the
dimension [c] = L−1
M−1
but according to the scaling ansatz (spatial
or temporal) we have [c] = L−2
. It confirms that one can use mass or
size interchangeably to characterize particles in the system. Let us define
dN is the number of particles per unit length (number density in higher
dimension) then it is connected to c(x, t) via
dN = c(x, t)dx, (3.57)
where dN has the dimension L−1
in the case of one dimensional system.
So, the number concentration N in the system is defined as
N =
Z ∞
0
c(x, t)dx, (3.58)
and hence for uniform distribution of particles [N] = L−1
. This is called
the zeroth moment of the distribution function c(x, t). Let us assume
that dM is the total length or size of the particles having size range
[x, x + dx] per unit mass and it is defined as
dM = xc(x, t)dx. (3.59)
The above relation implies that it bears no dimension for homogeneous
distribution of particles and hence it remains constant for all time. Thus
the total size or mass density M is
M =
Z ∞
0
xc(x, t)dx, (3.60)
which is the first moment of the distribution function is a conserved
quantity. In general, the nth
moment of the distribution function is de-
fined as
Mn(t) =
Z ∞
0
xn
c(x, t)dx, (3.61)
such that M0(t) = N(t) and M1(t) = M(t).
The question is: How can we manipulate data obtained from numerical
simulation to find the distribution function c(x, t)? It is possible to plot
a histogram of the number frequency normalized by the total mass and
class width. The first step would be to divide the number frequency into
classes of suitable widths. That is, we find how many particles are there](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-93-320.jpg)
![86 Stochastic theory of fragmentation
within the class [xi, xi +∆x]. Say, there are ni particles within this class
which we normalize by the bin size or class width ∆x and then divide
it by total mass or size L of all the particles to obtain ni
L×∆x. Now the
area under such histogram will give the total number of particles per
unit mass and this is exactly the zeroth moment of c(x, t). Similarly,
if we multiply the area ni
L×∆x of each bar of the histogram by their
corresponding particle size xi and then find the sum over all the bars
will give the total mass or size per unit mass. This will essentially be
the first moment of the distribution function. In general,
P
i xn
i
ni
L×∆x is
equivalent to the nth moment of the distribution function c(x, t).
The variance of the distribution function is defined as
σ =
p
x2 − x 2, (3.62)
where
x2
=
R ∞
0 x2
c(x, t)dx
R ∞
0 c(x, t)dx
=
M2(t)
M0(t)
, (3.63)
and
x =
R ∞
0
xc(x, t)dx
R ∞
0 c(x, t)dx
=
M1(t)
M0(t)
. (3.64)
The fragmentation equation that we have derived is mean-field in char-
acter since fluctuation is ignored altogether (σ = 0 as we shall see latter).
3.3 Model I: Sum kernel: F(x, y) = (x + y)λ−1
A model for the theory of fragmentation within the rate equation ap-
proach is specified by an explicit choice of the fragmentation kernel with
respect to its argument. We first solve the fragmentation equation for
the simplest sum kernel
F(x, y) = (x + y)λ−1
. (3.65)
Incorporating it in the fragmentation equation yields
∂c(x, t)
∂t
+ xλ
c(x, t) = 2
Z ∞
x
yλ−1
c(y, t)dy. (3.66)
For further discussion and analysis we find it convenient to incorpo-
rate the definition of the nth moment Mn(t) =
R ∞
0
xn
c(x, t)dx into rate](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-94-320.jpg)
![88 Stochastic theory of fragmentation
that can be written in terms of Kumer’s function
Mn(t) = 1F1
n − 1
λ
;
n + 1
λ
; −t
. (3.75)
In the long limit t → ∞ we find that
M(t) ∼ t− n−1
λ . (3.76)
One can arrive at the same solution for the nth moment by assuming
that
Mn(t) ∼ A(n)tα(n)
. (3.77)
Using it in Eq. (3.67) and insisting dimensional consistency gives
α(n + λ) = α(n) + 1. (3.78)
Iterating it subject to the condition that α(1) = 0 required by the con-
servation law gives
α(n) = −
n − 1
λ
, (3.79)
which gives the same asymptotic temporal solution as Eq. (3.76).
We thus find that the n moment of the distribution function exhibits
power-law with exponent linear with the exponent linear in n which is
the hall mark for simple scaling. It also exemplifies a robust feature of the
system in the sense that one can define the mean particle size as s(t) =
Mn(t)/Mn−1(t) instead of defining it strictly as s(t) = M1(t)/M0(t).
That is, regardless of the precise value of index n, the same scaling
exponent describes the asymptotic value of s(t). Using Eq. (??) in the
definition of the mean particle size we find that
s(t) ∼ t− 1
λ . (3.80)
It clearly reveals that the mean particle size is a decreasing quantity in
time which is in fact one of the characteristic feature of the fragmen-
tation process. We shall now do a simple dimensional analysis to the
rate equation. Note that according to Eq. (3.66) the distribution func-
tion c(x, t) is the governed parameter while x and t are the governing
parameters of the process i.e.,
c = c(x, t). (3.81)
On the other the dimension of the coefficient of c(x, t) on the left hand
side of the rate equation are ∂
∂t and xλ
and they must bear dimen-
sion T −1
revealing the dimensional relation [x] = [t− 1
λ ]. Note that this](https://image.slidesharecdn.com/nesmbookkhm-260822092903-d508b559/85/Comprehensive-Study-of-Statistical-Mechanics-in-Non-Equilibrium-Phenomena-96-320.jpg)
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Comprehensive Study of Statistical Mechanics in Non-Equilibrium Phenomena
Explores scaling, stochastic processes, fragmentation, fractals, complex networks, and phase transitions in statistical mechanics of non-equilibrium systems.
