Explicit Solution of the Generalised Langevin Equation
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Explicit Solution of the Generalised Langevin Equation Ivan Di Terlizzi1,2 · Felix Ritort3 · Marco Baiesi1,2 Received: 5 May 2020 / Accepted: 17 September 2020 © The Author(s) 2020
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. Keywords Stochastic dynamics · Fluctuations · Entropy production · Memory effects
1 Introduction The driven diffusion process of a colloidal particle or bead immersed in a fluid has become a paradigm of nonequilibrium physics [1–7]. Fluctuations play a prominent role for this mesoscopic system due to the multitude of random hits on the particle by the molecules of the surrounding fluid. If these molecules are tinier and faster than the colloidal particle, a net separation of timescales between fast and slow degrees of freedom occurs and the colloidal particle undergoes Markovian dynamics. In this case, the motion of the particle can be equivalently described by using the Langevin equation, path integrals and the FokkerPlank equation [8]. Historically, the Langevin approach came first and arguably remains the most intuitive. In fact, for a one dimensional system, by incorporating the effects of the fluid in Newton’s second law one may write a Langevin equation of motion for the position x(t)
Communicated by Abhishek Dhar.
B
Marco Baiesi [email protected]
1
Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padua, Italy
2
INFN, Sezione di Padova, Via Marzolo 8, 35131 Padua, Italy
3
Condensed Matter Physics Department, University of Barcelona, C/Marti i Franques s/n, 08028 Barcelona, Spain
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I. Di Terlizzi et al.
of a particle of mass m as a second order stochastic differential equation, ˙ + F (x, t) + ξ(t) . m x(t) ¨ = −γ0 x(t)
(1)
The random force is generated by a Gaussian white noise ξ(t), with average ξ(t) = 0 and correlation ξ(t )ξ(t ) = 2γ0 k B T δ(t − t ). The prefactor of the delta function ensures thermodynamic consistency according to the (second) fluctuation-dissipation theorem [9], linking the drag coefficient γ0 of the dissipative term −γ0 x˙ to the strength of the noisy term. As a deterministic force not due to the fluid we focus on the case F (x, t) = −∂x U (x, t) with a t
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