Advances in High Energy Physics

Volume 2019, Article ID 2635917, 27 pages

https://doi.org/10.1155/2019/2635917

## Steady States, Thermal Physics, and Holography

Theory Division, Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India

Correspondence should be addressed to Arnab Kundu; moc.liamg@udnuk.banra

Received 16 November 2018; Accepted 20 January 2019; Published 6 March 2019

Guest Editor: Cynthia Keeler

Copyright © 2019 Arnab Kundu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

It is well known that a Rindler observer measures a nontrivial energy flux, resulting in a thermal description in an otherwise Minkowski vacuum. For systems consisting of large number of degrees of freedom, it is natural to isolate a small subset of them and engineer a steady state configuration in which these degrees of freedom act as Rindler observers. In Holography, this idea has been explored in various contexts, specifically in exploring the strongly coupled dynamics of a fundamental matter sector, in the background of adjoint matters. In this article, we briefly review some features of this physics, ranging from the basic description of such configurations in terms of strings and branes, to observable effects of this effective thermal description.

#### 1. Introduction

Thermodynamics is ubiquitous. Typically, for a collection of large number of degrees of freedom, be it strongly interacting or weakly interacting, a thermodynamic description generally holds across various energy scales and irrespective of whether it is classical or quantum mechanical. The underlying assumption here is a notion of at least a* local thermal equilibrium*, for which such a formulation is possible. Intuitively, this is simple to define: a thermal equilibrium occurs when there is no net flow of energy. Typically this can be characterized by an intensive variable, temperature, with zero or very small time variation. The* smallness* needs to be established in terms of the smallest time-scale that is present in the corresponding system.

While thermodynamics has a remarkable reach of validity, equilibrium is still an approximate description of Nature, at best. Most natural events are dynamical in character. Of these, a particular class of phenomena can be easily factored out, that of systems at steady state. While steady state systems are not strictly in thermodynamic equilibrium, they can be described in terms of stationary macroscopic variables. For such systems, there is a nonvanishing expectation value of a flow, such as an energy flow or a current flow, which does not evolve with time. Typically, such states can be reached* asymptotically* starting from a generic initial state, or they appear as* transient* states before time evolution begins.

In this article, we will consider a similar situation. The prototype will consist of a* bath* degree of freedom, which is assumed to be infinitely large and will serve the purpose of a reservoir. In this* bath* background, we will consider the dynamics of a* probe* sector. In this sector, a stationary configuration can be easily constructed, by dumping all the excess energy into the reservoir. For example, consider a nonvanishing current flow in the probe sector. There will be work done to maintain the constant current flow; therefore it is expected that the actual description is dynamical. However, if we engineer the bath sector as a source of providing this energy, or a sink in which this energy is deposited, the resulting configuration remains stationary.

In the framework of quantum field theory (QFT), a similar construction was considered by Feynman-Vernon in [1], based on the Schwinger-Keldysh formalism of [2, 3]. For some review on this formalism, see,* e.g.,* [4–6]. In recent times, much work has gone into the reformulation of the Schwinger-Keldysh formalism; see,* e.g.,* [7–9]. While much of our subsequent discussion potentially has a leg deep inside this formalism, we will not make explicit use of the formalism, to keep a terse discourse. The basic idea is based on the so-called thermofield double construct, which has appeared long time back in [10]. Subsequently, in [11], a connection of classical black holes with the thermofield double construct was also established. For us, these two ideas are sufficient.

Consider the basic idea behind the thermofield double. Consider a quantum mechanical system, with a Hamiltonian and a complete set of eigenstates , such thatwhere is the energy of the corresponding eigenstate. Evidently, constitute a basis of the Hilbert space. Let us now double the total degrees of freedom, by considering two copies of the same system. At the level of the dynamics, these two copies of degrees of freedom are noninteracting between them. (The total Hamiltonian of the doubled system can be defined as or . With the latter choice, the thermofield double state does not evolve with time.) Therefore, the Hilbert space of the doubled quantum system is spanned by . Given this, the thermofield double state is defined asThis is certainly a special state in the doubled quantum system. We can assign a density matrix corresponding to this state: . This is a pure density matrix, as can be explicitly checked by establishing .

Given such a pure density matrix, let us compute the reduced density matrix while integrating out one copy of the system. Thus we obtainThe reduced density matrix, denoted above by , appears thermal in nature, with a temperature . Thus, given the thermofield double state, one can construct an equivalent thermal description. The process of integrating out one copy of the system may be conducted in various ways: this can be thought of as integrating out a subsystem to compute entanglement between the two. In the context of a black hole, similar to [11], or in the presence of a causal horizon, one can construct a Kruskal extension of the geometry. This maximal extension of the geometry can be thought of as the thermofield double and by integrating out one side, an effective thermal density matrix is obtained. It is also clear from the above discussion that, given any gauge invariant observable or a collection of such operators acting on the untraced system, denoted by , the expectation value is simply given bywhich is the thermal expectation value.

A similar picture holds true in the Holographic framework, which will be the primary premise in our subsequent discussions. In [12], eternal black holes in an asymptotically AdS geometry were proposed to be dual to two copies of the conformal field theory (CFT). Each CFT corresponds to the dual CFT that is defined on the conformal boundary of AdS. The basic picture is represented in Figure 1. In the Euclidean signature, the corresponding thermofield double state is created by the Euclidean path integral over an interval of . The thermofield double state, defined in (2), is maximally entangled from the point of view of the doubled degrees of freedom. Tracing over one copy produces a thermal effective description and this seems to lie at the core of the construction. Motivated by this, one can surmise that a qualitative emergent description of a thermofield double state ensures an effective thermodynamics. For this to happen, the essential ingredient is a black hole like causal structure. An intriguing idea relating quantum entanglement and the existence of an Einstein-Rosen bridge has recently been proposed in [13].