Fluctuation symmetries for work and heat

Abstract

We consider a particle dragged through a medium at constant temperature as described by a Langevin equation with a time-dependent potential. The time dependence is specified by an external protocol. We give conditions on potential and protocol under which the fluctuations of the dissipative work satisfy an exact symmetry for all times. We also present counterexamples to that fluctuation theorem when our conditions are not satisfied. Finally, we consider the dissipated heat, which differs from the work by a temporal boundary term. We explain why there is a correction to the standard fluctuation theorem due to the unboundedness of that temporal boundary. However, the corrected fluctuation relation has again a general validity.

Heat fluctuations for underdamped Langevin dynamics

“…One can check that this identity is verified by all linear diffusion systems studied so far, both theoretically [2,7,8,[14][15][16][17] and experimentally [18,26], although it was unnoticed [27].An example is given in the Supplemental Material [24].…”

“…At first sight, µ A (λ) should be the same function for the whole class [A] Qi of observables that differ from Q i by only temporal boundary terms. However, [2] and subsequent work [6][7][8][9][10][11][12][13][14][15][16][17] showed that this is not always true. To see this, it is convenient to rewrite the characteristic function for A ∈ [A] Qi under the alternative form…”

“…Note that β is not generically a pole of Z ∆U (λ, t), the characteristic function of ∆U [24]. In other words, Q = W − ∆U cannot be treated as the sum of two independent random variables for very large t, with the poles attributed to ∆U, as is often done [8,11,19,40].…”

“…This gives rise to exponential tails in the pdf of Q and is signaled by the presence of singularities in the corresponding characteristic function. Such temporal "boundary" effects that take place in systems with unbounded potentials are now well-documented, both theoretically [6][7][8][9][10][11][12][13][14][15][16][17] and experimentally [18][19][20].…”

Large deviation function and fluctuation theorem for classical particle transport

“…A well documented example is that of Brownian particle in contact with one or several heat reservoirs [5][6][7][8][9]. The large deviation function probes exponentially unlikely events for the heat evacuated to the reservoirs.…”