Explicit Solution of the Generalised Langevin Equation

Abstract

Generating an initial condition for a Langevin equation with memory is a non trivial issue. We introduce a generalisation of the Laplace transform as a useful tool for solving this problem, in which a limit procedure may send the extension of memory effects to arbitrary times in the past. This method allows us to compute average position, work, their variances and the entropy production rate of a particle dragged in a complex fluid by an harmonic potential, which could represent the effect of moving optical tweezers. For initial conditions in equilibrium we generalise the results by van Zon and Cohen, finding the variance of the work for generic protocols of the trap. In addition, we study a particle dragged for a long time captured in an optical trap with constant velocity in a steady state. Our formulas open the door to thermodynamic uncertainty relations in systems with memory.

Memory-induced oscillations of a driven particle in a dissipative correlated medium

“…of equation ( 55) -the present one involves a second term ∝ C q (t) due to thermal fluctuations. Note that the integration in the variable u in equation ( 54) runs from −∞, and this fact prevents a direct solution of the equation of motion by using the Laplace transform [50]. However, in order to determine the response of the average particle position to a sudden displacement X 0 imposed at time t = 0 from its stationary value X = 0, one can look for a solution X(t) of equation ( 55) with X(t) ≡ 0 for t < 0, and X(t) = X 0 at t = 0.…”

Work Extraction and Performance of Colloidal Heat Engines in Viscoelastic Baths

“…They involve the variance of the particle position at an arbitrary time t ≥ 0, x(t) 2 , with t = 0 the time defining the initial condition, computed over an ensemble of independent realizations of the colored noise ζ (t) defined by Equations ( 2). An analytical treatment of this problem requires the explicit solution of the generalized Langevin Equation ( 8), which is not trivial even in the simpler case of a constant trap stiffness and constant temperature [48]. Therefore, to address the problem of the performance of a Brownian Stirling heat engine described by Equations ( 2), ( 8), ( 13) and ( 14), we opt for numerical simulations of the corresponding stochastic dynamics.…”

“…Although all these conditions are met by a colloidal heat engine operating in a complex fluid, to the best of our knowledge they have never been examined in the context of stochastic thermodynamic cycles. Therefore, it is of paramount importance to assess the role of viscoelasticity in the performance of this kind of engines, since the resulting frequency-dependent friction experienced by a colloidal particle can significantly impact the rate at which energy is dissipated into a viscoelastic bath [45][46][47][48].…”

Work extraction and performance of colloidal heat engines in viscoelastic baths

“…They involve the variance of the particle position at an arbitrary time t, x(t) 2 , computed over an ensemble of independent realizations of the colored nosie ζ(t) defined by Equations ( 2). An analytical treatment of this problem requires the explicit solution of the generalized Langevin equation ( 8), which is not trivial even in the simpler case of a constant trap stiffness [45]. Therefore, to address the problem of the performance of a Brownian Stirling heat engine described by Equations ( 2), ( 8), ( 11) and ( 12), we opt for numerical simulations of the corresponding stochastic dynamics.…”

“…Although all these conditions are met by a colloidal heat engine operating in a complex fluid, to the best of our knowledge they have never been examined in in the context of stochastic thermodynamic cycles. Therefore, it is of paramount importance to assess the role of viscoelasticity in the performance of this kind of engines, since the resulting frequency-dependent friction experienced by a colloidal particle can significantly impact the rate at which energy is dissipated into a viscoelastic bath [42,43,44,45].…”