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Exactly solvable nonlinear wave equations -- colloquially known as the soliton systems -- are widely regarded as one of the greatest achievements of mathematical physics. But somehow, aside of several mathematical frameworks and other formal aspects, not a lot seems to be known about statistical properties of classical integrable field theories and, in particular, classical transport properties at finite temperature. In this talk, we present a kinetic equation to deal with dense soliton gases, expressed in terms of a linear integral dressing equation for the soliton spectral function which accounts for renormalization of the soliton velocities due to the interactions with a soliton many-body state. This is accomplished in the framework of the algebro-geometric integration technique which permits to classify all quasi-periodic solutions of an integrable equation of motion in terms of the moduli of finite-genus Riemann surfaces. By identifying soliton excitations, applying Born-Sommerfeld quantization for soliton orbits, extracting the two-body S-matrix, and finally taking the thermodynamic finite-density, the free energy functional is expressed as the saddle point of the action-space path-integral. Our hydrodynamic equations can understood as the thermodynamic analogue of the celebrated Whitham's modulation equations. The equations are universal, and even apply to the quantum theories of solitons. If time permits, we show how to obtain a closed compact formula for the Drude weight in the quantum Heisenberg spin chain, and discuss peculiarities related to it.

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Enej Ilievski, University of Amsterdam
Seminar Room 0.03, ETP
Contact: Zala Lenarcic