This is the summary of a study we conducted to simulate heat transfer in one dimension of same and alternating mass systems using statistical mechanics and molecular dynamics.

1. Simulation and Study of
Heat Transfer
Keerthana P. G.

2. Why should we study heat transfer?
 Insulation and radiant barriers
 Heat exchangers
used in refrigeration, air conditioning, space heating, power generation, and chemical
processing.
 Heat sinks also help to cool electronic and optoelectronic devices such as
CPUs, higher-power lasers, and light-emitting diodes (LEDs).
 In order to understand these materials and their uses, it is necessary to
understand their mechanical properties such as stress coefficients, thermal
conductivity etc.
 Our project is an attempt to verify several theoretical predictions made
regarding Hard Sphere models in 1D system

3. What this is all about.
 To simulate heat conduction in a 1-dimensional chain with different atoms to
verify theoretical predictions
 To study one dimensional heat conduction and verify its properties.
 The long term objective is to study heat transfer and investigate the
influence of the working fluid on a finite time Carnot’s Engine.

4. Isolated 3d system
 FCC lattice of 4*8*8*8 argon atoms
 Defining the lattice
 Initialising positions and momenta of all atoms.
 Positions by forced initialisation
 Velocities using Marsaglia Bray method.
.

5. Algorithm

6.  Forces calculation from lennard jones potential
 Periodic Boundary Condition with minimum image condition
 Velocity verlet algorithm to update forces and positions

7. Stabilization

8. Heat Conduction in 1D Chain of Atoms
 N particles in a 1D box of length L
 Forced position initialisation.
 Velocity from Maxwell-Boltzmann distribution in compliance with the initial
system temperature.
 The left and right hand walls are kept at two different temperatures, in our
case, 8K and 2K respectively.
 Atoms have a small but finite radius.

9.  Interactions:
1. Particle-particle interactions by hard collisions. We assume that all
interactions are elastic.
2. Interaction with the wall: The atoms at the end, upon hitting the wall
bound back with a velocity sampled from Rayleigh distribution.
Here, TL is the temperature of the wall from which the atom is bounding back.

10.  Local temperatures are calculated according the expression described.
 Temperature profile is obtained at steady state.
 BBGKY(Boboligov-Born-Green-Kirkwood-Yvon) Equations to model the system.

11. Theoretical Prediction
 Surprisingly, the temperature profiles in the case of equal masses and the one
with arbitrarily small mass differences completely different.
 But energy density is constant in space at steady state. Seemingly
contradicting.
 Temperature profile doesn’t change under m(i)-->c*m(i)
 Also, from the boundary conditions, it can be verified that T(cT1, cT2,
x)=cT(T1, T2, x)
Explanation: Temperature also depends on n(x), i.e. the local number
density.
T(x,t)= 2 ε(x,t)/n(x,t)

12. 1D system(with 1 particle type)
 N particles of same mass in a 1D box of length L
 Collisions are assumed to be elastic.
 Initialisation is done at a certain temperature,the left and right hand walls
are kept at 3K and 2K respectively.
 Temperature is calculated for every individual atom and a temperature
profile is obtained.

13. Temperature profile
 Theoritical prediction says that the temperature profile will be flat with
themperature 𝑇1 𝑇2 .Here is the simulated temperature profile

14. Energy profile

15. 1D system with Particles of Alternating
Mass
 N particles in a tube of length L. Alternate particles have alternate masses.
 The ratio of masses (m1/m2) is varied to get different temperature profiles.
 Once again we assume elastic collisions.

16. Temperature profile
Expected profile

17. Observations:
 Temperature has a smooth and continuously varying profile with jumps at the
boundaries that tend to smoothen with increase in system size.
 For small =(𝑚2 − 𝑚1)/𝑚1 and large N , the temperature profile depends on
and only by a scaling factor of

18. Energy Profile
The energy density profile here is supposed to be the same as the
profile for a same mass system

19. The Way Forward
 Study of the effects of working fluid on the performance of a finite time
Carnot’s cycle.

20. Acknowlegments:
 Physics Review Letter, 2001, Heat Conduction in a One-Dimensional Gas of
Elastically Colliding Particles of Unequal Mass, Abhishek Dhar, Raman Research
Institute, Bangalore
 Research Article, 2008, Heat Transfer in Low Dimensional systems, Abhishek
Dhar, Raman Research Institute, Bangalore

21. Thank You