 02481 Mark Pollicott and Howard Weiss
 Free Eenergy as a Dynamical and Geometric Invariant (or Can You Hear
the Shape of a Potential?)
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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 onedimensional
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\'etype 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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