Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change

Details Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change

2011-12-14

arxiv.org/abs/1112.3075v2

Physics

Condensed Matter

Statistical Mechanics

18 pages, 1 Figure

Scientific paper

This work extends a recently developed mathematical theory of thermodynamics for Markov processes with, and more importantly, without detailed balance. We show that the Legendre transform in connection to ensemble changes in Gibbs' statistical mechanics can be derived from the stochastic theory. We consider the joint probability $p_{XY}$ of two random variables X and Y and the conditional probability $p_{X|Y=y^*}$, with $y^*=$ according to $p_{XY}$. The stochastic free energies of the XY system (fluctuating Y ensemble) and the $X|Y$ system (fixed Y ensemble) are related by the chain rule for relative entropy. In the thermodynamic limit, defined as $V,Y\to\infty$ (where one assumes Y as an extensive quantity, while V denotes the system size parameter), the marginal probability obeys $p_Y(y)\to \exp(-V\phi(z))$ with $z=y/V$. A conjugate variable $\mu=-\phi(z)/z$ naturally emerges from this result. The stochastic free energies of the fluctuating and fixed ensembles are then related by $F_{XY}=F_{X|Y=y^*}-\mu y^*$, with $\mu=\partial F_{X|Y=y}/\partial y$. The time evolutions for the two free energies are the same: $d[F_{XY}(t)]/dt=\partial [F_{X|Y=y^*(t)}(t)]/ \partial t$. This mathematical theory is applied to systems with fixed and fluctuating number of identical independent random variables (idea gas), as well as to microcanonical systems with uniform stationary probability.

Affiliated with

Also associated with

No associations

Rating

Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-225046