# Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics

@inproceedings{Ruelle2004ThermodynamicFT, title={Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics}, author={David Ruelle}, year={2004} }

1. Introduction to the 2nd edition 2. Introduction 3. Theory of Gibbs States 4. Gibbs States: complements 5. Translation invariance: theory of equilibrium states 6. Connection between Gibbs States and equilibrium 7. One-dimensional systems 8. Extension of the thermodynamic formalism Appendix A.1. Miscellaneous definitions and results Appendix A.2. Topological dynamics Appendix A.3. Convexity Appendix A.4. Measures and abstract dynamical systems Appendix A.5. Integral representations on convex…

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## References

SHOWING 1-10 OF 64 REFERENCES

Statistical Mechanics: Rigorous Results

- Physics
- 1999

Thermodynamic behaviour - ensembles the thermodynamic limit for thermodynamic functions - lattice systems the thermodynamic limit for thermodynamic functions - continuous systems low density…

Mean entropy of states in classical statistical mechanics

- Mathematics
- 1967

The equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. We consider here continuous and lattice systems and define a mean entropy…

A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule

- Physics
- 1967

It is shown that for an infinite lattice system, thermodynamic equilibrium is the solution of a variational problem involving a mean entropy of states introduced earlier [2]. As an application, a…

Statistical mechanics of quantum spin systems. III

- Mathematics
- 1968

AbstractIn the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space
$$\mathfrak{B}$$
of translationally…

Statistical mechanics of a one-dimensional lattice gas

- Mathematics, Physics
- 1968

We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phase…

Two remarks on extremal equilibrium states

- Economics
- 1973

First it is shown that each extremal equilibrium state is representable as limit of Gibbs states in finite volumes, and that an analogous statement holds for extremal invariant equilibrium states.…

A Relativised Variational Principle for Continuous Transformations

- Mathematics
- 1977

The formula (1.1) also follows from (1.2). Actually we prove a more general relative variational principle by considering pressure instead of entropy, and this result generalises the variational…

Observables at infinity and states with short range correlations in statistical mechanics

- Economics
- 1968

We say that a representation of an algebra of local observables has short-range correlations if any observable which can be measured outside all bounded sets is a multiple of the identity, and that a…

A heuristic theory of phase transitions

- Mathematics
- 1977

LetZ be a suitable Banach space of interactions for a lattice spin system. Ifn+1 thermodynamic phases coexist for Φ0 ∈Z, it is shown that a manifold of codimensionn of coexistence of (at least)n+1…