Extended heat-fluctuation theorems for a system with deterministic and stochastic forces

R. van Zon and E. G. D. Cohen

Phys. Rev. E 69, 056121 – Published 28 May 2004

Abstract

Heat fluctuations over a time $τ$ in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution of these fluctuations for all $τ$ . By a saddle-point method we obtain analytical results for the inverse Fourier transform, which, for not too small $τ$ , agree very well with numerical results from a sampling method as well as from the fast Fourier transform algorithm. Due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation theorem is, for infinite and for finite $τ$ , replaced by an extended fluctuation theorem that differs noticeably and measurably from it. In particular, for large fluctuations, the ratio of the probability for absorption of heat (by the particle from the fluid) to the probability to supply heat (by the particle to the fluid) is much larger here than in the conventional fluctuation theorem.

Received 14 November 2003

DOI:https://doi.org/10.1103/PhysRevE.69.056121

Authors & Affiliations

R. van Zon and E. G. D. Cohen

The Rockefeller University, 1230 York Avenue, New York, New York 10021, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 69, Iss. 5 — May 2004

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The PDFs $π_{j}^{Q}$ obtained by two numerical methods, for $w = 2.0$ and $τ = 1.0$ . Sampling method results ( $N_{s} = 2 000 000$ , $δ p = 0.2$ ) are shown as crosses for $π_{T}^{Q}$ and as dots for $π_{S}^{Q}$ , respectively. Fast Fourier transform results ( $δ q = 4 \times 10^{- 4}$ , $q_{m a x} = 200$ ) are shown as a thin dashed line for $π_{T}^{Q}$ and as a bold dashed line for $π_{S}^{Q}$ . The inset shows the corresponding fluctuation functions $f_{j}^{Q}$ [Eq. (44)] and a solid straight line indicating the prediction of the conventional FT.Reuse & Permissions
Figure 2
Numerical results for the fluctuation functions $f_{j}^{Q}$ for the stationary $(j = S)$ and the transient case $(j = T)$ , for (a) $w = 1, τ = 4$ , (b) $w = 2, τ = 2.5$ , (c) $w = 3, τ = 1.3333$ , and (d) $w = 4, τ = 1$ . Sampling method results $(N_{s} = 1.2 \times 10^{9}, δ p = 0.2)$ are shown as thin crosses for $f_{T}^{Q}$ and as bold dots for $f_{S}^{Q}$ . Fast fourier transform results $(δ q = 4 \times 10^{- 4}, q_{m a x} = 200)$ are shown as the dashed thin curves for $f_{T}^{Q}$ and as dashed bold curves for $f_{S}^{Q}$ . The solid straight line indicates the prediction of the conventional FT.Reuse & Permissions
Figure 3
Numerical results for the $τ$ dependence of the fluctuation functions $f_{j}^{Q}$ for $w = 2.0$ for (a) the transient $(j = T)$ and (b) the stationary case $(j = S)$ . Sampling results $(N_{s} = 1.2 \times 10^{9}, δ p = 0.2)$ are shown as symbols: $(*)$ $τ = 0.5$ , $(\circ)$ $τ = 1$ , $(•)$ $τ = 2$ , $(◻)$ $τ = 4$ . Fast Fourier transform results $(δ q = 4 \times 10^{- 4}, q_{m a x} = 200)$ are shown as the dashed curves (the $τ$ values of these curves are the same as those of the coinciding sampling points). The solid straight line is the conventional FT result.Reuse & Permissions
Figure 4
Geometry in the complex $q$ plane of the saddle points, singularities, and paths of steepest descent of $h (q)$ [Eqs. (48, 54)] for the stationary state $[Δ_{j} = 0]$ , for $w = 1$ , $τ = 4$ , and $p = - 2$ , $0$ , and $4.25$ , respectively. In all graphs, solid dots depict the position of the saddle points $A$ , $B$ , $C$ , and $D$ . Wiggly curves give the branch cuts ending in the branch points at $q = \pm i$ which are indicated with a thin horizontal line. The branch point at $- i$ is also a pole. The dashed line $R$ gives the original line of integration, i.e., the real axis. Solid lines labeled $S_{A}$ , $S_{B}$ , $S_{C}$ , and $S_{D}$ are paths of steepest descent though the saddle point, on which $Re h$ attains a maximum.Reuse & Permissions
Figure 5
Comparison of the saddle-point approximation with the numerically obtained solution, for $w = 0.5$ and (a) $τ = 2.5$ and (b) $τ = 5$ . The solid thin line is the transient PDF $π_{T}^{Q} (p; τ)$ as calculated using the saddle-point approximation of Sec. 4 and the dashed thin curve is the same PDF calculated using the numerical inverse Fourier transform of Sec. 3 $(δ q = 2 \cdot 10^{- 4}, q_{m a x} = 2^{18} δ q = 52.4288)$ . The solid bold line is the stationary PDF $π_{S}^{Q} (p; τ)$ calculated using the saddle-point approximation, and the bold dashed curve is that same PDF obtained using the numerical inverse Fourier transform.Reuse & Permissions
Figure 6
Results for the large $τ$ behavior of (a) $f_{T}^{Q} (p; τ)$ and (b) $f_{S}^{Q} (p; τ)$ for $w = 1$ , as found from the saddle-point approximation. The figures show four lines corresponding to the result for $τ = 3$ , $τ = 10$ , $τ = 100$ , and $τ = 1000$ , as well as a solid black straight line $(p)$ that gives the behavior expected from the conventional FT.Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics