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Entropy and the Arrow of Time

Entropy and the Arrow of Time. T h. T c. Q h. Q c. Christian Van den Broeck Hasselt University. Second law. bead. bead. DNA handles. Q. RNA strand. Small scale?. our work W. P(W). micro- pipette. water. heat bath T. <W> ≥ F. laser trap. Reverse Process.

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Entropy and the Arrow of Time

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  1. Entropy and the Arrow of Time Th Tc Qh Qc Christian Van den Broeck Hasselt University

  2. Second law bead bead DNA handles Q RNA strand Small scale? our work W P(W) micro- pipette water heat bath T <W>≥ F laser trap

  3. Reverse Process Forward Process V V Crooks PRE 60, 2721 (1999) ~ P(-W) Jarzynski C. PRL 78, 2690 (1997) P(W) P(W) ~ <W>≥ F ≥ <-W> F Collin, D. et al. Nature 437, 231 (2005) Cleuren VdB Kawai PRL 96 , 050601 (2006).

  4. Hamiltonian calculation of <W> itself? work source heat bath T W H() system : A -> B Canonical Equilibrium T lB Canonical Equilibrium T lA ~ What is W? Reverse experiment : B -> A W <W>≥ F=FB-FA

  5. equilibrium t0 t0 t t t1 t1 equilibrium - - - What is W? work source Work performed along given trajectory? W H(q,p;) system : A -> B Liouville Reversibility

  6. t0 t0 t t t1 t1 equilibrium - - - Dissipation equilibrium - - - - Trajectory dependent dissipated heat. Entropy production = relative entropy phase space densities

  7. Beautifull:properties relative entropy! Second law: inequality replaced by equality. Requires full statistical information. Valid independent of distance from equilibrium. (valid for any intermediate time t)

  8. Deep meaning: Stein’s lemma The entropy production is equal to the ease for identifying the arrow of time.

  9. Wdis=D()>0 Wdis=D()=0 ~ ~ . . . . . . atypical realization! How (un)likely are the typical forward realizations in the time reverse experiment?

  10. partial info no info Useful: chain rule Relative entropy decreases upon coarse graining Second law replaced by stronger inequality as more information becomes available.

  11. measurement k Wdis τ  50 bins 10 bins Illustration: Brownian particle in harmonic potential in thermal bath Gomez-Marin Parrondo, Van den Broeck PRE 78, 011107 (2008).

  12. Path formulation Gomez-Marin Parrondo, Van den Broeck EPL 82, 50002, (2008). Relative entropy unchanged upon addition dependent variables Microscopic path: redundant information. However: this expression allows to prune at the level of paths. Correct expression for stochastic processes. Jiu-Li Van den Broeck Nicolis, Z. Phys B56, 165 (1984) Maes Netocny, J Stat Phys 110, 269 (2003) Gaspard, J Stat Phys 117, 599 (2004) Seifert, Phys Rev Lett 94, 040602 (2005) Jarzynski, Phys Rev E 73, 046105 (2006)

  13. T Conditional dissipation bounded from below but no longer by F Szilard engine Computation

  14. Conditional dissipation bounded from below but no longer by F Computation - Landauer principle

  15. Multi-canonical Constrained equilibrium Grand-Canonical (Micro-Canonical) Parrondo Van den Broeck Kawai arXiv: dissipation relaxation

  16. Path formulation Gomez-Marin Parrondo, Van den Broeck arXiv:0710.4290 Crooks G.E. G.E. Crooks, PRE 60, 2721 (1999) B. Cleuren, R. Kawai, C. Van den Broeck, PRL (2006) , 2721 (1999) Correct description of work implies correct description of dissipation!

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