Entropy and the Arrow of Time

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Entropy and the Arrow of Time
Christian Van den Broeck Hasselt
University
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P(W)
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Jarzynski C. PRL 78, 2690 (1997)
P(W)
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Hamiltonian calculation of ltWgt itself?
work source
W
Canonical Equilibrium T lA
Canonical Equilibrium T lB

What is W? Reverse experiment ? ?B -gt ?A W
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What is W?
Work performed along given trajectory?
Liouville
Reversibility
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Dissipation
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Trajectory dependent dissipated heat.
Entropy production relative entropy phase
space densities
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Beautifullproperties relative entropy!
Second law inequality replaced by
equality. Requires full statistical
information. Valid independent of distance from
equilibrium. (valid for any intermediate time t)
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Deep meaning Steins lemma
The entropy production is equal to the ease for
identifying the arrow of time.
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How (un)likely are the typical forward
realizations in the time reverse experiment?
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Useful chain rule
Relative entropy decreases upon coarse graining
Second law replaced by stronger inequality as
more information becomes available.
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Illustration Brownian particle in harmonic
potential in thermal bath
Gomez-Marin Parrondo, Van den Broeck PRE 78,
011107 (2008).
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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)
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Computation
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Computation
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Landauer principle
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Multi-canonical Constrained equilibrium Grand-Cano
nical (Micro-Canonical)
Parrondo Van den Broeck Kawai arXiv
dissipation relaxation
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Path formulation
Gomez-Marin Parrondo, Van den Broeck
arXiv0710.4290
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!