The distribution of heat fluctuations in resistively-coupled dual temperature heat baths

Abstract

A path integral approach used earlier to calculate the exact heat distribution function of a harmonically trapped Brownian oscillator at a fixed temperature (Chatterjee and Cherayil 2010 Phys. Rev. E 82 051104) is extended in this paper to a consideration of heat fluctuations in a dual temperature system. Our new calculations complement recent experimental data obtained by Ciliberto et al (2013 J. Stat. Mech. P12014) on the stochastic thermodynamics of an electrical circuit made up of coupled resistors maintained at two distinct temperatures. Measurements of various thermodynamic quantities in this system, including the heat, work and energy, reveal trends that represent interesting generalizations of results for the single temperature case. In particular, the measured distribution of the heat exchanged at one of the reservoirs is found to agree qualitatively with a fluctuation relation applicable at long times. In the present work, we exploit the formal equivalence between the electrical circuit and a system of coupled Brownian oscillators to derive, within the path integral formalism and for a special set of parameter values, an exact integral representation-valid for all times-of the total heat distribution function of the system. We find that the infinite-time limit of this distribution shows interesting departures from its expected behavior.

Fluctuation relations for flow-driven trapped colloids and implications for related polymeric systems

“…But now, because of the path dependence of W, the probability density P (x, y, t|x 0 , y 0 ) that appears in equation ( 8) must be formulated in path-dependent terms too. This is done by following the same steps used in earlier derivations from this lab of related probability density functions for other model systems [37][38][39]. Without going into the details of those derivations, suffice it to say that P (x, y, t|x 0 , y 0 ) can be shown to be given by…”

“…Being a quadratic functional of x (t) and y(t), the propagator G can be expressed exactly in the form G = φ(t) exp(− Ā), where φ(t) is a proportionality constant that ensures that ∞ −∞ dx ∞ −∞ dyP (x, y, t|x 0 , y 0 ) is 1, and Ā, the classical action, is the value of A ≡ t 0 dt L evaluated along its classical trajectory, which is obtained from the solution to the Euler-Lagrange equations (d/dt)∂L/∂ ẋ = ∂L/∂x and (d/dt)∂L/∂ ẏ = ∂L/∂y under the boundary conditions x(t) = x, x(0) = x 0 , y(t) = y and y(0) = y 0 [40,41]. One can show that the functions φ(t) and Ā are given, respectively, by [38,39]…”

Heat Distribution of Relativistic Brownian Motion

“…Heat is a fundamental quantity in Stochastic Thermodynamics, i.e., the energy naturally exchanged between the system and the surrounding, in a disordered way. As a random variable, characterization of the statistics of heat for diffusive systems was carried in many different models [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. These works bring physical insights into the thermodynamics of classical diffusive systems, however, as far as we known, they only deals with nonrelativistic systems.…”

Heat fluctuations in the logarithm-harmonic potential

“…Characterization of the statistics of the thermodynamic quantities of diffusive systems was carried in many different models. Derivations of the distribution and characteristic function for work and heat had been solved for Brownian systems, ranging from the free particle to non-harmonic potentials with theoretical [5][6][7][8][9][10][11][12][13][14][15][16] and experimental [17][18][19][20] results. These works bring physical insights in the thermodynamics of such systems.…”