Abstract
We propose a general formalism, within large-deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the “extended fluctuation relations” of Bernard and Doyon follow from the linearity of the hydrodynamic equations, forming a marker of “freeness” much like the absence of hydrodynamic diffusion does. We show how an extension of the formalism gives exact exponential behaviours of spatio-temporal two-point functions of twist fields, with applications to order-parameter dynamical correlations in arbitrary homogeneous, stationary state. We explain in what situations the large-deviation principle at the basis of the results fail, and discuss how this connects with nonlinear fluctuating hydrodynamics. Applying the formalism to conformal hydrodynamics, we evaluate the exact cumulants of energy transport in quantum critical systems of arbitrary dimension at low but nonzero temperatures, observing a phase transition for Lorentz boosts at the sound velocity.
Highlights
We propose a general formalism, within large-deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states
We show how to calculate the scaled cumulant generating function for transport of any conserved quantity in stationary, homogeneous, clustering states of many-body systems, in or out of equilibrium
The technique is based on large-deviation theory, and the result is expressed in terms of quantities readily available from the Euler hydrodynamics description of the system
Summary
Far-from-equilibrium physics has seen a large amount of theoretical and experimental developments in recent years [1,2,3,4,5]. Free-particle advanced techniques have been used [15,16,17,18,19,20] (see [21]), and exact results exist in certain integrable impurity models [22] and in general 1 + 1-dimensional conformal field theory (CFT) [23,24] (see the review [25]) In these studies, non-equilibrium currents are generated by the partitioning protocol [26,27,28] (see [25] and references therein). The theory is based on biasing the measure by a total time-integrated current, accessing rare fluctuations and explicitly generating the scaled cumulants Such a bias is a widely used technique in stochastic systems (sometimes referred to as exponential tilting, or s-ensemble), and how it gives rise to a new stochastic dynamics is referred to as the (classical or quantum) generalised Doob transform (see [52,53] and references therein). In “Appendix A” we provide the main derivation of the general results; in “Appendix B”, we review basic aspects of Euler hydrodynamics, including the solution to the Riemann problem in free (linear) hydrodynamics and the normal modes of conformal hydrodynamics; and in “Appendix C”, we discuss the multi-parameter SCGF and present related general arguments
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