Generalized hydrodynamics and approach to Generalized Gibbs equilibrium for a classical harmonic chain
arxiv(2024)
摘要
We study the evolution of a classical harmonic chain with nearest-neighbor
interactions starting from domain wall initial conditions. The initial state is
taken to be either a product of two Gibbs Ensembles (GEs) with unequal
temperatures on the two halves of the chain or a product of two Generalized
Gibbs Ensembles (GGEs) with different parameters in the two halves. For this
system, we construct the Wigner function and demonstrate that its evolution
defines the Generalized Hydrodynamics (GHD) describing the evolution of the
conserved quantities. We solve the GHD for both finite and infinite chains and
compute the evolution of conserved densities and currents. For a finite chain
with fixed boundaries, we show that these quantities relax as ∼ 1/√(t)
to their respective steady-state values given by the final expected GE or GGE
state, depending on the initial conditions. Exact expressions for the Lagrange
multipliers of the final expected GGE state are obtained in terms of the steady
state densities. In the case of an infinite chain, we find that the conserved
densities and currents at any finite time exhibit ballistic scaling while, at
infinite time, any finite segment of the system can be described by a
current-carrying non-equilibrium steady state (NESS). We compute the scaling
functions analytically and show that the relaxation to the NESS occurs as ∼
1/t for the densities and as ∼ 1/t^2 for the currents. We compare the
analytic results from hydrodynamics with those from exact microscopic numerics
and find excellent agreement.
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