Thermodynamic formula for the cumulant generating function of time-averaged current

Abstract

The cumulant generating function of time-averaged current is studied from an operational viewpoint. Specifically, for interacting Brownian particles under non-equilibrium conditions, we show that the first derivative of the cumulant generating function is equal to the expectation value of the current in a modified system with an extra force added, where the modified system is characterized by a variational principle. The formula reminds us of Einstein's fluctuation theory in equilibrium statistical mechanics. Furthermore, since the formula leads to the fluctuation-dissipation relation when the linear response regime is focused on, it is regarded as an extension of the linear response theory to that valid beyond the linear response regime. The formula is also related to previously known theories such as the Donsker-Varadhan theory, the additivity principle, and the least dissipation principle, but it is not derived from them. Examples of its application are presented for a driven Brownian particle on a ring subject to a periodic potential.

Large deviations of the current for driven periodic diffusions

“…This SDE represents one of the simplest nonequilibrium system violating detailed balance for γ = 0 and has played, as such, an important role in the development and illustration of recent results about nonequilibrium response [1][2][3][4], entropy production [5][6][7], and large deviations in the longtime [8][9][10][11][12] or low-noise [13][14][15] regime. It is also used as a model of Josephson junctions subjected to thermal noise [16][17][18], Brownian ratchets [19], and manipulated Brownian particles [20][21][22], among other systems (see [1]), and is thus an ideal experimental testbed for the physics of nonequilibrium systems.…”

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

“…In that case, the equations of motion (32) for the auxiliary process coincide with the non-equilibrium system (20), with f ext = sT. However, as noted above, the heat flow in the auxiliary process is given by (7).…”

“…We emphasise that this case V 0 = 0 is a special one-if one inspects the statistics of the particle trajectories (that is, (q t , p t ) t∈[0,τ] ) then it is not possible to determine whether one is observing the non-equilibrium system (20) or the conditioned equilibrium system based on (19). On the other hand, if one observes (by some physical measurement) the heat flow into the reservoir for these two cases, then one sees that the conditioned process has no dissipation (no net heat flow), but the non-equilibrium process does have a finite rate of heat flow into the environment.…”

“…At the level of sample paths, b =b. We identify γ(∂G s /∂p i ) as a "control force" [21,22] that realises the required flux J (see also [20]). …”

“…The conditioned distribution P J is not easy to analyse. To make further progress, it is useful to define an "auxiliary" Markov process [16][17][18][19][20][21] whose steady state path measure is close to P J : see [16] for a comprehensive discussion.…”