Minimal dissipation with viscoelastic baths in weakly driven processes

TL;DR

The paper addresses thermodynamics of overdamped systems driven weakly while coupled to viscoelastic baths. It develops a generalized Langevin framework with memory, establishing that thermodynamic consistency requires an added Dirac delta in the friction kernel and introducing the persistence time $\tau_P$, then analyzes relaxation via linear response, deriving the relaxation function $\Psi_0(t)$ and relaxation time $\tau_R$. Through two canonical protocols (moving-trap and stiffening-trap) it shows that memory can either be inert or accelerate relaxation, notably reducing $\tau_R^B$ for the stiffening trap and lowering irreversible work $W_{\rm irr}$; it also derives near-optimal driving protocols $g^*(t)$ whose form depends on $\tau_P$ via coefficients $a_n$. The results highlight that memory effects imprint measurable thermodynamic signatures and can be exploited to improve energetic efficiency in complex environments, with broader relevance to soft matter and biological systems. Data and code are provided in the associated repository.

Abstract

We investigate the thermodynamics of overdamped systems weakly driven by time-dependent protocols while interacting with viscoelastic heat baths. Using a generalized Langevin equation with memory, we derive the conditions under which the friction kernel ensures thermodynamic consistency, notably requiring the addition of a Dirac delta. Within linear response theory, we compute the relaxation function and relaxation time for two classes of protocols: moving and stiffening harmonic traps. Surprisingly, we find that viscoelastic memory does not always hinder relaxation; in certain cases, it accelerates it by reducing the effective relaxation time, leading to lower dissipation. We also derive optimal protocols that minimize the irreversible work and show how they are modified by the presence of the persistence time of the viscoelastic heat bath. Our results reveal that memory effects in the overdamped regime leave measurable thermodynamic signatures, depending on the protocol, with direct implications for controlling complex systems.

Figures (3)

• Figure 1: Fourier transform of the relaxation function of the stiffening trap in VHB. The function is positive, corroborating that the system is compatible with the Second Law of Thermodynamics. The unity was used for all parameters.
• Figure 2: Comparison for the linear driving of the stiffening trap case for $\tau_P=0$ and $\tau_P=1$. The last one has less irreversible work than the first case, corroborating that the system is closer to the near-to-equilibrium case. It was used $\gamma=10$, $\omega_0=10$. The unity of the irreversible work is $\Psi_0(0)\delta\lambda^2$.
• Figure 3: Minimal irreversible work for moving laser trap (panels (a), (b), (c)) and stiffening trap (panels (d), (e), (f)). In all cases, the continuous linear approximation of Eq. \ref{['eq:nopt']} produces a lesser irreversible work than the linear case of Eq. \ref{['eq:linear']}. For each case, the persistence time $\tau_P=0.1,1,10$ were used, corroborating no influence on the optimality. The values $\gamma=10$, $\omega_0=10$, $\beta=1$ were used. The unity of the irreversible work is $\Psi_0(0)\delta\lambda^2.$