[Paper Review] Fluctuation relations without microscopic time reversality: Generalized Green-Kubo relation and integral fluctuation theorem for uniformly sheared granular systems

Song‐Ho Chong, Michio Otsuki+1 more|ArXiv.org|Jun 10, 2009

Advanced Thermodynamics and Statistical Mechanics4 references3 citations

TL;DR

This paper derives a generalized Green-Kubo relation and an integral fluctuation theorem for uniformly sheared granular systems without relying on microscopic time-reversal symmetry, which is broken by inelastic collisions. The key contribution is an exact expression for steady-state properties in terms of transient time-correlation functions, enabling theoretical analysis of nonequilibrium steady states in dissipative systems like granular matter and jammed glasses.

ABSTRACT

We derive the generalized Green-Kubo relation and an integral form of the fluctuation theorem that apply to uniformly sheared granular systems in which microscopic time-reversal symmetry is broken. The former relation provides an exact representation of nonequilibrium steady-state properties as the integral of the time-correlation function describing transient dynamics from an initial quiescent towards a final sheared steady state. We also investigate implications of the integral fluctuation theorem on the approach towards the steady state and on the possible form of the steady-state distribution function in terms of the excess thermodynamic function.

Motivation & Objective

• To extend fluctuation theorems and Green-Kubo relations to granular systems where microscopic time-reversal symmetry is broken due to inelastic collisions.
• To establish a theoretical framework for nonequilibrium steady states in macroscopic dissipative systems such as sheared granular matter.
• To derive an exact expression for steady-state properties using transient time-correlation functions from a quiescent initial state to a sheared steady state.
• To investigate the form of the steady-state distribution function in terms of excess thermodynamic quantities without assuming time-reversal symmetry.
• To provide a foundation applicable to jammed glassy materials where time-reversal symmetry is also absent.

Proposed method

• Uses the SLLOD equations of motion to model uniformly sheared granular systems with inelastic hard-sphere interactions.
• Defines the Liouville operator and phase-space dynamics for systems with broken time-reversal symmetry due to dissipative forces.
• Derives an integral fluctuation theorem by analyzing forward and backward path probabilities in phase space, leading to a symmetry in the exponential of the excess entropy production.
• Applies cumulant expansion techniques to the path probability ratio to express the steady-state distribution in terms of excess thermodynamic functions.
• Introduces a small parameter $\epsilon \sim \dot{\gamma} \sim \gamma$ to analyze the $\epsilon \to 0$ limit and derive corrections to the steady-state distribution.
• Uses the generalized Green-Kubo relation to express steady-state properties as the integral of time-correlation functions from an initial quiescent state to the final sheared steady state.

Experimental results

Research questions

• RQ1 Can the generalized Green-Kubo relation be derived without assuming microscopic time-reversal symmetry in granular systems?
• RQ2 Does an integral form of the fluctuation theorem hold in systems with broken time-reversal symmetry, such as uniformly sheared granular matter?
• RQ3 How can the steady-state distribution function be expressed in terms of excess thermodynamic quantities when time-reversal symmetry is absent?
• RQ4 What is the role of transient dynamics in determining steady-state properties in the absence of time-reversal invariance?
• RQ5 Can the derived formalism be applied to jammed glassy materials where time-reversal symmetry is also broken?

Key findings

• The generalized Green-Kubo relation is derived as an exact expression for steady-state properties in terms of the time-correlation function of transient dynamics from an initial quiescent state to a final sheared steady state.
• An integral fluctuation theorem is established that relates the probabilities of forward and backward paths in phase space, even without microscopic time-reversal symmetry.
• The steady-state distribution function is shown to be expressible in terms of the excess entropy production, extending previous results that relied on time-reversal symmetry.
• The correction terms in the distribution function expansion are found to be $O(\epsilon^3 t^2)$, indicating that the leading-order expression is valid when $\epsilon t \ll 1$.
• The formalism is applicable to jammed glassy materials, suggesting broader relevance beyond granular systems.
• The derivation confirms that the steady-state properties can be accessed via transient dynamics, even in the absence of time-reversal symmetry, by using path probability symmetries in the excess entropy production.