Equilibrium States, Phase Transitions and Dynamics in Quantum Anharmonic Crystals

  • Yuri KozitskyEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)


The basic elements of the mathematical theory of states of thermal equilibrium of infinite systems of quantum anharmonic oscillators (quantum crystals) are outlined. The main concept of this theory is to describe the states of finite portions of the whole system (local states) in terms of stochastically positive KMS systems and path measures. The global states are constructed as Gibbs path measures satisfying the corresponding DLR equation. The multiplicity of such measures is then treated as the existence of phase transitions. This effect can be established by analyzing the properties of the Matsubara functions corresponding to the global states. The equilibrium dynamics of finite subsystems can also be described by means of these functions. Then three basic results of this theory are presented and discussed: (a) a sufficient condition for a phase transition to occur at some temperature; (b) a sufficient condition for the suppression of phase transitions at all temperatures (quantum stabilization); (c) a statement showing how the phase transition can affect the local equilibrium dynamics.


KMS state Path measure Stochastic process Green function 


82B10 82B20 82B26 



The author was supported by the DFG through the SFB 701 “Spektrale Strukturen and Topologische Methoden in der Mathematik” and by the European Commission under the project STREVCOMS PIRSES-2013-612669.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsMaria Curie-Skłodowska UniversityLublinPoland

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