Thermodynamics of polarized relativistic matter
HJE
Thermodynamics of polarized relativistic matter
Pavel Kovtun 0 1 2
0 PO Box 1700 STN CSC, Victoria BC, V8W 2Y2 , Canada
1 Department of Physics and Astronomy, University of Victoria
2 @ T =T is the acceleration
We give the free energy of equilibrium relativistic matter subject to external gravitational and electromagnetic elds, to one-derivative order in the gradients of the external elds. The free energy allows for a straightforward derivation of bound currents and bound momenta in equilibrium. At leading order, the energy-momentum tensor admits a simple expression in terms of the polarization tensor. Beyond the leading order, electric and magnetic polarization vectors are intrinsically ambiguous. The physical e ects of polarization, such as the correlation between the magneto-vortically induced surface charge and the electro-vortically induced surface current, are not ambiguous.
Holography and quark-gluon plasmas; Thermal Field Theory
1 Introduction 2 Free energy 3
Leading order in the derivative expansion
Next order in the derivative expansion
Weak electromagnetic elds Strong electromagnetic elds: 1+1 dimensions Strong electromagnetic elds: 2+1 dimensions Strong electromagnetic elds: 3+1 dimensions
5
Summary
1
Introduction
We would like to understand collective macroscopic behaviour of matter subject to external
elds. In the high-temperature limit this is often captured by classical hydrodynamics. The
ingredients that go into writing down the hydrodynamic equations are: the identi cation of
relevant variables (conserved densities, order parameters), the derivative expansion (small
gradients near equilibrium), and symmetry constraints. The hydrodynamic equations are
modi ed when the system is subject to external electric and magnetic elds. The latter will
induce polarization (electric, magnetic, or both) in a
uid, and as a result the transport
properties of the uid will change. Our focus here will be on isotropic relativistic matter
because a) electromagnetic
elds are intrinsically relativistic, b) relativistic
uids have
more symmetry than non-relativistic uids, and c) relativistic uids have been a subject of
much recent attention in the literature due to their appearance in heavy-ion physics [1, 2],
in gravitational physics, through the holographic duality [3], and even in condensed matter
cedure for obtaining the energy-momentum tensor and the current density for stationary
equilibrium polarized matter subject to external gravitational and electromagnetic elds.
We will nd simple expressions for \bound" currents, including equilibrium surface currents
and surface momenta.
Let us start with the standard description of equilibrium thermodynamics without
external elds. In the grand canonical ensemble at temperature T0 = 1= 0 and chemical
potential 0, extensivity in the large-volume limit dictates that the logarithm of the
grandcanonical partition function Z[T0; 0] is proportional to the d-dimensional spatial volume,
0, see e.g. [7]. In the path integral action, the fundamental elds of the
microscopic theory can then be coupled to time-independent external sources: the
(Euclidean) metric gE and the (Euclidean) gauge eld AE. The gauge eld couples to the
conserved current, whose time component is the charge density corresponding to the
chemical potential. See ref. [8] for a convenient parametrization of the Euclidean sources gE and
AE. The Euclidean path integral gives rise to the partition function Z = Z[T0; 0; gE; AE],
where T0 = 1= 0 is the coordinate periodicity of the Euclidean time. We assume that the
coupling to time-independent external sources leaves the system in equilibrium, so that no
entropy is produced. The temperature and the chemical potential will be altered by the
external sources and are not uniform any more. For example, the equilibrium temperature
becomes T (x) = T0=pg0E0(x) [6]. Similarly, the chemical potential will be shifted by the
time component of the external gauge eld. We can write W =
i ln Z as
W [T0; 0; gE; AE] = i 0
Z
ddx pgE F (T0; 0; gE; AE) ;
(1.1)
where pgE is the square root of the determinant of gE , and F is the negative of the grand
canonical free energy density. In at space and without external gauge elds, F reduces to
the pressure P , and in general F is a complicated function of the spatially varying external
sources. In a slight abuse of terminology, we will refer to F as the free energy density,
and to W as the free energy. Varying W with respect to a time-independent source gives
rise to a zero-frequency insertion in the Euclidean path integral of the operator coupled
to the source. The relevant operators are the energy-momentum tensor (coupled to the
metric), and the conserved current (coupled to the gauge eld). Thus W is the generating
functional for zero-frequency correlation functions of the energy-momentum tensor and the
current in equilibrium.
The Euclidean external sources gE and AE may be \un-Wick-rotated" to Minkowski
time to obtain the (...truncated)