Effective Temperature in Steady-state Dynamics from Holography

TL;DR

This paper demonstrates that in certain gauge-gravity dual systems under steady-state conditions, an effective temperature and free energy can be defined for probe degrees of freedom, despite the absence of a well-defined entropy.

Contribution

It introduces a holographic framework for defining an effective temperature in steady-state systems with probe branes, using an open string equivalence principle and analyzing the probe-sector's fluctuations.

Findings

Effective temperature is uniquely determined by the open string metric.

The framework applies to various stringy examples with steady electric fields.

The effective temperature depends on the fluctuation sector in non-Abelian cases.

Abstract

We argue that, within the realm of gauge-gravity duality, for a large class of systems in a steady-state there exists an effective thermodynamic description. This description comes equipped with an effective temperature and a free energy, but no well-defined notion of entropy. Such systems are described by probe degrees of freedom propagating in a much larger background, e.g. $N_f$ number of ${\cal N} =2$ hypermultiplets in ${\cal N}=4$ $SU(N_c)$ super Yang-Mills theory, in the limit $N_f \ll N_c$. The steady-state is induced by exciting an external electric field that couples to the hypermultiplets and drives a constant current. With various stringy examples, we demonstrate that an open string equivalence principle determines a unique effective temperature for all fluctuations in the probe-sector. We further discuss various properties of the corresponding open string metric that determines the effective geometry which the probe degrees of freedom are coupled to. We also comment on the non-Abelian generalization, where the effective temperature depends on the corresponding sector of the fluctuation modes.