belousov.tel

DPG Spring meeting of the Condensed Matter Section 2026

Microcanonical ensemble out of equilibrium

Roman Belousov with J. Elliott, F. Berger, L. Rondoni, and A. Erzberger

Erzberger Lab | EMBL, Heidelberg

Boltzmann's monode

Microcanonical ensemble: all states are equally likely, energy $E$ and number of particles $N$ are fixed.

Boltzmann entropy: $S_{\rm B}(\text{macrostate}) = k_{\rm B} \ln W(\text{microstates} | \text{macrostate})$
Euler equation for equilibrium thermodynamics: $E = T S - P V + \mu N$
Statistics of fluctuations: $p(\text{macrostate}) \propto e^{S_{\rm B}(\text{macrostate})/k_{\rm B}}$

Gibbs' ensemble theory

Probability and information theory
(Shannon-)Gibbs entropy: $S_{\rm G} = - k_{\rm B} \int_{\text{microstates}} p(\text{microstate})\ \ln p(\text{microstate})$
Jaynes' MaxEnt: $p(\text{microstate}) = \arg\max_{p}\ S_{\rm G}[p\ |\ \text{constraints}]$

Jaynes' caliber theory

Caliber (path entropy): $S_{\rm J} = - \int_{\text{paths}} \ p(\text{path})\ \ln p(\text{path})$
Jaynes' MaxCal: $p(\text{path}) = \arg\max_{p}\ S_{\rm J}[p\ |\ \text{constraints}]$

How to choose the constraints?

Fluctuation theorems (constant average flux)?: How to apply such constraints to experimental systems (e.g. in biology)?; What about systems with gradients (e.g. of matter) not reproduced by the constraint of a constant average flux?; What about noise at the short time scales (e.g. fluctuation-dissipation theorem for Langevin models)?

How to choose the constraints?

Fluctuation theorems (constant average flux)?: How to apply such constraints to experimental systems (e.g. in biology)?; What about systems with gradients (e.g. of matter) not reproduced by the constraint of a constant average flux?; What about noise at the short time scales (e.g. fluctuation-dissipation theorem for Langevin models)?