21,287 research outputs found
Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
The approach to the theory of many-particle interacting systems from a
unified standpoint, based on the variational principle for free energy is
reviewed. A systematic discussion is given of the approximate free energies of
complex statistical systems. The analysis is centered around the variational
principle of N. N. Bogoliubov for free energy in the context of its
applications to various problems of statistical mechanics and condensed matter
physics. The review presents a terse discussion of selected works carried out
over the past few decades on the theory of many-particle interacting systems in
terms of the variational inequalities. It is the purpose of this paper to
discuss some of the general principles which form the mathematical background
to this approach, and to establish a connection of the variational technique
with other methods, such as the method of the mean (or self-consistent) field
in the many-body problem, in which the effect of all the other particles on any
given particle is approximated by a single averaged effect, thus reducing a
many-body problem to a single-body problem. The method is illustrated by
applying it to various systems of many-particle interacting systems, such as
Ising and Heisenberg models, superconducting and superfluid systems, strongly
correlated systems, etc. It seems likely that these technical advances in the
many-body problem will be useful in suggesting new methods for treating and
understanding many-particle interacting systems. This work proposes a new,
general and pedagogical presentation, intended both for those who are
interested in basic aspects, and for those who are interested in concrete
applications.Comment: 60 pages, Refs.25
Continuous-time integral dynamics for Aggregative Game equilibrium seeking
In this paper, we consider continuous-time semi-decentralized dynamics for
the equilibrium computation in a class of aggregative games. Specifically, we
propose a scheme where decentralized projected-gradient dynamics are driven by
an integral control law. To prove global exponential convergence of the
proposed dynamics to an aggregative equilibrium, we adopt a quadratic Lyapunov
function argument. We derive a sufficient condition for global convergence that
we position within the recent literature on aggregative games, and in
particular we show that it improves on established results
Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity
This paper presents a pure complementary energy variational method for
solving anti-plane shear problem in finite elasticity. Based on the canonical
duality-triality theory developed by the author, the nonlinear/nonconex partial
differential equation for the large deformation problem is converted into an
algebraic equation in dual space, which can, in principle, be solved to obtain
a complete set of stress solutions. Therefore, a general analytical solution
form of the deformation is obtained subjected to a compatibility condition.
Applications are illustrated by examples with both convex and nonconvex stored
strain energies governed by quadratic-exponential and power-law material
models, respectively. Results show that the nonconvex variational problem could
have multiple solutions at each material point, the complementary gap function
and the triality theory can be used to identify both global and local extremal
solutions, while the popular (poly-, quasi-, and rank-one) convexities provide
only local minimal criteria, the Legendre-Hadamard condition does not guarantee
uniqueness of solutions. This paper demonstrates again that the pure
complementary energy principle and the triality theory play important roles in
finite deformation theory and nonconvex analysis.Comment: 23 pages, 4 figures. Mathematics and Mechanics of Solids, 201
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