Statistical mechanics of double sinh-Gordon kinks. (English) Zbl 0958.82037
Summary: We study the classical thermodynamics of the double sinh-Gordon (DSHG) theory in \(1+1\) dimensions. This model theory has a double well potential \(V (\phi)= (\zeta \cosh 2\phi-n)^2\) when \(n> \zeta\), thus allowing for the existence of kinks and antikinks. Though it is non-integrable, the DSHG model is remarkably amenable to analysis. Below we obtain exact single kink and kink lattice solutions as well as the asymptotic kink-antikink interaction. In the continuum limit, finding the classical partition function is equivalent to solving for the ground state of a Schrödinger-like equation obtained via the transfer integral method. For the DSHG model, this equation turns out to be quasi-exactly solvable. We exploit this property to obtain exact energy eigenvalues and wavefunctions for several temperatures both above and below the symmetry breaking transition temperature (provided \(n= 1,2,\dots, 6)\). The availability of exact results provides an excellent testing ground for large scale Langevin simulations. The probability distribution function (PDF) calculated from Langevin dynamics is found to be in striking agreement with the exact PDF obtained from the ground state wavefunction. This validation points to the utility of a PDF-based computation of thermodynamics utilizing Langevin methods. In addition to the PDF, field-field and field fluctuation correlation functions were computed and also found to be in excellent agreement with the exact results.
MSC:
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
Keywords:
double sinh-Gordon equation; quasi-exactly solvable potential; kink lattice; probability distribution function; Langevin dynamicsReferences:
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