Kerchof Hall, Room 317
3:30, Friday, April 20 and Monday 23
Speaker: Vojkan Jaksic, McGill University
Title: An introduction to mathematical theory of non-equilibrium quantum statistical mechanics.
Abstract: In this mini-course I shall discuss
mathematical foundations of non-equilibrium quantum statistical mechanics focusing
on a class of recent developments
which fall roughly into two categories:
(A) Axiomatic results that concern mathematical structure of the theory;
(B) Study of concrete physically relevant models;
In the first lecture I will focus on (A) in the simple setting of finite dimensional
quantum systems. The exposition will be pedagogical and no background
beyond basic linear algebra is required. After introducing the terminology
(finite dimensional relative entropy, Gibbs canonical ensemble, free energy, etc),
I shall discuss entropy production observable and
entropy production balance equation, finite time linear response theory,
finite time fluctuation theorems and their relation to linear response.
In the first part of second lecture, which will be technically more advanced,
I shall discuss how these finite time results extend
to infinite dimensional systems (by either taking thermodynamic
limit of finite systems or by considering directly infinite dimensional
C^\ast-dynamical systems).
Taking time to infinity and showing that finite
time structural results hold for the limiting states (called non- equilibrium steady states)
and that the resulting relations are non-trivial
is a fundamental problem which in context of (A) is solved by making axiomatic
assumptions concerning ergodic properties of the system. The axioms
can be verified only in the context of concrete models and that leads
to (B) and some deep analytical problems. In the second part of the second lecture I will discuss
some concrete physically relevant models for which axioms of (A) can be verified and,
time permitting, some related open problems.
References:
[1] Aschbacher, W., Jaksic V., Pautrat, Y.,Pillet, C.-A.:
Topics in non-equilibrium quantum statistical
mechanics. In Open Quantum Systems III. S. Attal, A. Joye,
C.-A. Pillet editors. Lecture Notes in Mathematics 1882,
Springer, New York (2006).
[2] Jaksic, V.,Pillet, C-A.: On entropy production in quantum statistical mechanics.
Commun. Math. Phys. 217, 285 (2001).
[3] Jaksic, V., Pillet, C.-A.:
Mathematical theory of non-equilibrium quantum statistical mechanics.
J. Stat. Phys. 108, 787 (2002).
[4] Ruelle, D.: Natural nonequilibrium states in quantum statistical
mechanics. J. Stat. Phys. 98, 57 (2000).
[5] Ruelle, D.: Entropy production in quantum spin systems.
Commun. Math. Phys. 224, 3 (2001).
[6] Ruelle, D.: Topics in quantum statistical mechanics and operator
algebras. Preprint, mp-arc 01-257 (2001).