02-481 Mark Pollicott and Howard Weiss
Free Eenergy as a Dynamical and Geometric Invariant (or Can You Hear the Shape of a Potential?) (578K, pdf) Nov 22, 02
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Abstract. The lattice gas provides an important and illuminating family of models in statistical physics. An interaction $\Phi$ on a lattice $L \subset \Bbb Z^d$ determines an idealized lattice gas system with potential $A_\Phi$. The pressure $P(A_\Phi)$ and free energy $F_{A_\Phi}(\beta)= -(1/\beta) P(\beta A_\Phi)$ are fundamental characteristics of the system. However, even for the simplest lattice systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, potentials for certain model systems are determined by their free energy. In particular, we show that for a one-dimensional lattice gas, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a H\"older continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question Can you hear the shape of a drum?". We also introduce the free energy as a new geometric invariant for negatively curved surfaces and discuss some of its properties. In this case we show that the free energy is intimately related to a Poincar\'e-type series which encodes both the lengths of closed geodesics and word lengths of the corresponding words in the fundamental group. Thus free energy contains some refined information on the ratio of word length to hyperbolic length of closed geodesics, as studied by Milnor.

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