IDENTIFICATION OF THE POLARON MEASURE IN STRONG COUPLING AND THE PEKAR VARIATIONAL FORMULA

• 成果类型: Article
• 署名作者: Mukherjee, Chiranjib; Varadhan, S. R. S.
• 署名单位: University of Munster; New York University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1392
• 发表日期: 2020
• 页码: 2119-2144
• 关键词: brownian occupation measures; mean-field interaction

摘要

The path measure corresponding to the Frohlich polaron appearing in quantum statistical mechanics is defined as the tilted measure d (P) over cap (epsilon),T = 1/Z(epsilon, T) exp{1/2 integral-(T)(T)integral-(T)(T) epsilon e(-epsilon vertical bar t-s vertical bar)/vertical bar omega(t) - omega(s)vertical bar ds dt}dP. Here, epsilon > 0 is a constant known as the Kac parameter or the inverse-coupling parameter, and P is the distribution of the increments of the three-dimensional Brownian motion. In (Comm. Pure Appl. Math. 73 (2020) 350-383) it was shown that, when epsilon > 0 is sufficiently small or sufficiently large, the (thermodynamic) limit lim(T -> infinity) (P) over cap (epsilon,T) = (P) over cap (epsilon) exists as a process with stationary increments, and this limit was identified explicitly as a mixture of Gaussian processes. In the present article the strong coupling limit or the vanishing Kac parameter limit lim(epsilon -> 0) (P) over cap (epsilon) is investigated. It is shown that this limit exists and coincides with the increments of the so-called Pekar process, a stationary diffusion with generator 1/2 Delta + (del psi/psi).del, where psi is the unique (up to spatial translations) maximizer of the Pekar variational problem g(0) = sup(parallel to psi parallel to 2=1){integral(R3 )integral(R3) psi(2)(x)psi(2)(y)vertical bar x - y vertical bar(-1) dx dy -1/2 parallel to del psi parallel to(2)(2)}. As the Pekar process was also earlier shown (Ann. Probab. 44 (2016) 3934-3964; Ann. Inst. Henri Poincare Probab. Stat. 53 (2017) 2214-2228; Comm. Pure Appl. Math. 70 (2017) 1598-1629) to be the limiting object of the mean-field polaron measures, the present identification of the strong coupling limit is a rigorous justification of the mean-field approximation of the polaron problem (on the level of path measures) conjectured by Spohn in (Ann. Physics 175 (1987) 278-318). Replacing the Coulomb potential by continuous function vanishing at infinity and assuming uniqueness (modulo translations) of the relevant variational problem, our proof also shows that path measures coming from a Kac interaction of the above form with translation invariance in space converge to the increments of the corresponding meanfield model.