Integral fluctuation theorem

U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95, 040602 (2005). 

Various transient or steady-state forms of Eqn (15.50) are known as integral fluctuation theorems. Fluctuation theorems express universal properties of the probability distribution p Qi) for functionals Q[x(t)], like work, heat or entropy change, evaluated along the fluctuating trajectories taken from ensembles with well-specified... 

The convexity of the exponential functions then implies the inequality (Q) > 0, which resembles the second law. The integral fluctuation theorem implies that there are trajectories for which Q is negative with the exception of the degenerate case, p(Q) = d(Q) leading to violation of the second law. One constraint on the probability distribution p(Q). If p(Q) is a Gaussian, the integral flucmation theorem implies the relation ((Q — (O)) ) = 2(Q) between the variance and the mean of Q (Seifert, 2012). 

Here, 7 is the friction coefficient and Si is a Gaussian random force uncorrelated in time satisfying the fluctuation dissipation theorem, (Si(0)S (t)) = 2mrykBT6(t) [21], where 6(t) is the Dirac delta function. The random force is thought to stem from fast and uncorrelated collisions of the particle with solvent atoms. The above equation of motion, often used to describe the dynamics of particles immersed in a solvent, can be solved numerically in small time steps, a procedure called Brownian dynamics [22], Each Brownian dynamics step consists of a deterministic part depending on the force derived from the potential energy and a random displacement SqR caused by the integrated effect of the random force... 

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

The bath correlation function that relates to the spectral density via the fluctuation-dissipation theorem [Eq. (2.11)] can now be obtained via the contour integration algorithm. We have (setting + 7 )... 

Equating the integrants on both sides yields a second form of the fluctuation-dissipation theorem... 

Subtracting this result from (1.388) yields a similar equation for v, thus the mean and fluctuating velocities separately satisfy the continuity equation. This is a trivial consequence of the linearity of (1.388). In order to treat the equation of motion in the same way, we apply the Reynolds decomposition procedure on the instantaneous velocity and pressure variables in (1.391) and average term by term. It can be shown by use of Leibnitz theorem that the operation of time averaging commutes with the operation of differentiating with respect to time when the limits of integration are constant [15, 107, 122, 155]. 

The subscript FP signifies that only contributions from the flow and plateau regions should be taken into account in the integration. This equation is based on the fluctuation-dissipation theorem [42 ]. A relationship closely related to Eq. 5.12 makes use of complex compliance data. 

---