Fluctuation Theorem for Currents and Schnakenberg Network Theory

Abstract

A fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state, by using Schnakenberg network theory. The theorem can be applied, in particular, in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution.

Nonequilibrium version of the Einstein relation

“…(4), but with an effective value of s that is given by Eq. (6). Consequently the associated relation v/D = s is consistent with the exact analysis, while the bare relation v/D = s is violated.…”

Stochastic thermodynamics under coarse graining

“…The first term in Eq. (53) has the exact same form as the entropy production at the microstates level (22), and the second term is an ensemble average over the mesostates probabilities, P k , of the entropy production arising from within each mesostate. We now turn to the situation described in Sec.…”

“…Early developments in this field were restricted to the ensemble-averaged level and focused on steady-state situations [4][5][6][7][8]. The crucial conceptual breakthrough came later and consisted in identifying the central thermodynamic quantities at the level of single stochastic trajectories [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. The discovery of fluctuation theorems has played a major role in this regard [29][30][31][32][33].…”

The McLennan–Zubarev steady state distribution and fluctuation theorems

“…[4], it was formulated as an extension of Onsager's reciprocity relation. Furthermore, the fluctuation theorem is not restricted to dynamical systems, and was also confirmed for the Langevin system [5], for general stochastic systems [6], and for the master equation [10,13]. In ref.…”

“…To elucidate the nature of these fluctuations is an issue of nonequilibrium statistical physics in last two decades, for instance, in refs. [1,2,3,4,5,6,7,8,9,10,11,12,13].…”