Full statistics of nonstationary heat transfer in the Kipnis–Marchioro–Presutti model
Eldad Bettelheim, Naftali R. Smith, Baruch Meerson
Abstract
Abstract We investigate non-stationary heat transfer in the Kipnis–Marchioro–Presutti (KMP) lattice gas model at long times in one dimension when starting from a localized heat distribution. At large scales this initial condition can be described as a delta-function, u ( x , t = 0) = Wδ ( x ). We characterize the process by the heat transferred to the right of a specified point x = X by time T , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mo>∫</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> </mml:math> and study the full probability distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">P</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . The particular case of X = 0 has been recently solved by Bettelheim et al (2022 Phys. Rev. Lett. 128 130602). At fixed J , the distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">P</mml:mi> </mml:math> as a function of X and T has the same long-time dynamical scaling properties as the position of a tracer in a single-file diffusion. Here we evaluate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">P</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> by exploiting the recently uncovered complete integrability of the equations of the macroscopic fluctuation theory (MFT) for the KMP model and using the Zakharov–Shabat inverse scattering method. We also discuss asymptotics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">P</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> which we extract from the exact solution and also obtain by applying two different perturbation methods directly to the MFT equations.