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.

References in corpus (19)

• Stochastic thermodynamics, fluctuation theorems, and molecular machines
• Thermodynamic uncertainty relation for biomolecular processes
• Dissipation bounds all steady-state current fluctuations
• Proof of the Finite-Time Thermodynamic Uncertainty Relation for Steady-State Currents
• Finite-time generalization of the thermodynamic uncertainty relation
• Stationary and Transient Work-Fluctuation Theorems for a Dragged Brownian Particle
• Current fluctuations and transport efficiency for general Langevin systems
• Kinetic uncertainty relation
• Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms
• Non-equilibrium work fluctuations for oscillators in non-Markovian baths
• Oscillating Modes of Driven Colloids in Overdamped Systems
• Jarzynski Relation, Fluctuation Theorems, and Stochastic Thermodynamics for Non-Markovian Processes
• Frenetic bounds on the entropy production
• Fluctuation symmetries for work and heat
• Uncertainty relation under information measurement and feedback control
• Weak Galilean invariance as a selection principle for coarse-grained diffusive models
• Thermodynamic uncertainty relation for time-delayed Langevin systems
• Uncertainty relations for time-delayed Langevin systems
• A thermodynamic uncertainty relation for a system with memory

Cited by in corpus (8)

• Work extraction and performance of colloidal heat engines in viscoelastic baths
• Memory-induced oscillations of a driven particle in a dissipative correlated medium
• A thermodynamic uncertainty relation for a system with memory
• On the emergence of memory in equilibrium versus non-equilibrium systems
• Hyperballistic transport in dense systems of charged particles under ac electric fields
• Variance sum rule: proofs and solvable models
• The rise and fall of branching: a slowing down mechanism in relaxing wormlike micellar networks
• Observation of Super-ballistic Brownian Motion in Liquid