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Entropy Produce Molecule Motor

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- 1. Entropy production along a stochastic trajectory of a small system Joseph X. Zhou Biological Physics group MPI-PKS Jam session, June 6, 2008
- 2. <ul><li>In the time of modern physics about BEC, HT super conductor, cold atom, laser </li></ul>Why talking about Thermodynamics ? Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you any more. - Arnold Sommerfeld <ul><li>Thermodynamics laws need to be generalized </li></ul><ul><li>Do molecular machines work differently from those in macro-world </li></ul><ul><li>A complete thermodynamics laws include non-equilibrium </li></ul>When we move to small world… <ul><li>Is thermodynamics an outdated subject like a dinosaur ? </li></ul>
- 3. Thermodynamics development history <ul><li>1820 – 1850, First law and second law of thermodynamics </li></ul><ul><li>1900, Equilibrium state, </li></ul><ul><li>1930 – 1960, Non-equilibrium, Linear response, Onsager, Green-Kubo </li></ul><ul><li>> 1993, Non-equilibrium, Fluctuation Theorem, Jarzynski Equality </li></ul>
- 4. Thermodynamic systems characterized by length scales
- 5. Energy conserves - why machines have different efficiency <ul><li>Entropy production is different, and it decides the energy efficiency </li></ul><ul><li>All thermal machines are below Carnot cycle efficiency </li></ul><ul><li>Car – 25 %, </li></ul><ul><li>Power station – 46% </li></ul>
- 6. Molecular machine’s efficiency <ul><li>Low Reynolds-number world is very sticky </li></ul><ul><li>All kinetic energy are dissipated </li></ul><ul><li>Molecular machine’s efficiency should be low </li></ul><ul><li>Molecular machine’s efficiency is quite high </li></ul><ul><li>Kinesin – 60% </li></ul><ul><li>Ion pump on membrane – 70% </li></ul>J. Liphardt et. al., Science 296, 1832, 2002
- 7. Why molecular machine so efficient? <ul><li>Find your friend in a crowded dance floor </li></ul>- By random push from others - By pushing through the crowds - By taking random push when it is right direction
- 8. Stochastic thermodynamics <ul><li>First law on a trajectory </li></ul><ul><li>Longevin equation for overdamped system </li></ul><ul><li>On a single trajectory </li></ul><ul><li>So fluctuation dominate in small system, W, q no long definite number, P(W) and P(q) instead will reflect system characteristics. </li></ul>U. Seifert, J. Stat. Phy., 128, 2007
- 9. Fluctuation Theorem <ul><li>The corresponding Fokker-Planck eq. </li></ul><ul><li>Define entropy on the trajectory </li></ul><ul><li>Relationship to non-equilibrium ensemble entropy </li></ul><ul><li>From this definition, we can derive an integral fluctuation theorem </li></ul><ul><li>With concept of time-reverse trajectories, we can derive a more general relation: </li></ul>
- 10. Fluctuation Theorem and Jarzynski equality <ul><li>If we define entropy production rate , another form of fluctuation theorem can be written: </li></ul><ul><li>It implies that molecule machines may absorb heat to do work sometimes </li></ul><ul><li>Jarzynski equality can also be derived from generalized integral fluctuation theorem: </li></ul><ul><li>Compare with Equilibrium relationship: </li></ul>
- 11. Prospects of non-equilibrium thermodynamics <ul><li>For most biological molecular machines, JE doesn’t apply </li></ul><ul><li>Need to generalize JE to arbitrary transitions between nonequilibrium states </li></ul><ul><li>Quantum version of Fluctuation Theorem is also on its way </li></ul>C. Bustamante, Physics Today, 2005
- 12. Summary <ul><li>Nonequilibrium thermodynamics of small systems (biomolecular machines) is still a cutting-edge research field </li></ul><ul><li>The second thermodynamics law is generalized to the fluctuation theorem. Negative entropy is possible in the single trajectory. </li></ul><ul><li>Molecular machines take advantage of thermal noise in the micro-world to do work, which is the reason why they usually have a higher energy efficiency than their macro-counterparts. </li></ul><ul><li>Nonequilibrium thermodynamics still need to develop further to find the relationship of arbitrary transitions between Nonequilibrium states </li></ul>Acknowledgement: Thanks Prof. Udo Serfeit for the discussion and his numerous papers on Nonequ.

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