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Jean Bricmont

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Papers by Jean Bricmont

Non-translation invariant Gibbs states with coexisting phases: I. Existence of sharp interface for Widom-Rowlinson type lattice models in three dimensions

Communications in Mathematical Physics, 1979

We investigate the spatially inhomogeneous states of two component, A - B, Widom-Rowlinson type l... more We investigate the spatially inhomogeneous states of two component, A - B, Widom-Rowlinson type lattice systems. When the fugacity of the two components are equal and large, these systems can exist in two different homogeneous (translation invariant) pure phases one A-rich and one B-rich. We consider now the system in a box with boundaries favoring the segregation of these two phases into “top and bottom” parts of the box. Utilizing methods due to Dobrushin we prove the existence, in three or more dimensions, of a “sharp” interface for the system which persists in the limit of the size of the box going to infinity. We also give some background on rigorous results for the interface problem in Ising spin systems.

Lattice systems with a continuous symmetry. II. Decay of correlations

Communications in Mathematical Physics, 1981

We consider perturbations of a massless Gaussian lattice field on Z d , d ^ 3, which preserves th... more We consider perturbations of a massless Gaussian lattice field on Z d , d ^ 3, which preserves the continuous symmetry of the Hamiltonian, e.g., It is known that for all T > 0 the correlation functions in this model do not decay exponentially. We derive a power law upper bound for all (truncated) correlation functions. Our method is based on a combination of the log concavity inequalities of Brascamp and Lieb, reflection positivity and the Fortuin, Kasteleyn and Ginibre (F.K.G.) inequalities.

Lattice systems with a continuous symmetry. I. Perturbation theory for unbounded spins

Communications in Mathematical Physics, 1981

We consider perturbations of a massless Gaussian lattice field on Z d , d ^ 3, which preserves th... more We consider perturbations of a massless Gaussian lattice field on Z d , d ^ 3, which preserves the continuous symmetry of the Hamiltonian, e.g., It is known that for all T > 0 the correlation functions in this model do not decay exponentially. We derive a power law upper bound for all (truncated) correlation functions. Our method is based on a combination of the log concavity inequalities of Brascamp and Lieb, reflection positivity and the Fortuin, Kasteleyn and Ginibre (F.K.G.) inequalities.

Lattice systems with a continuous symmetry. III. Low temperature asymptotic expansion for the plane rotator model

Communications in Mathematical Physics, 1981

We consider perturbations of a massless Gaussian lattice field on Z d , d ^ 3, which preserves th... more We consider perturbations of a massless Gaussian lattice field on Z d , d ^ 3, which preserves the continuous symmetry of the Hamiltonian, e.g., It is known that for all T > 0 the correlation functions in this model do not decay exponentially. We derive a power law upper bound for all (truncated) correlation functions. Our method is based on a combination of the log concavity inequalities of Brascamp and Lieb, reflection positivity and the Fortuin, Kasteleyn and Ginibre (F.K.G.) inequalities.

Diffusion in Energy Conserving Coupled Maps

We consider a dynamical system consisting of subsystems indexed by a lattice. Each subsystem has ... more We consider a dynamical system consisting of subsystems indexed by a lattice. Each subsystem has one conserved degree of freedom ("energy") the rest being uniformly hyperbolic. The subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved. We prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit.

Rigorous renormalisation group and disordered systems

Physica A-statistical Mechanics and Its Applications, 1990

We consider random walks in a non-symmetric random environment. We report a recent result, based ... more

Reply to Gabriel Stolzenberg

Social Studies of Science, 2004

Determinism, Chaos and Quantum Mechanics

After some general remarks on the notion of "determinism", I will discuss the precise meaning of ... more After some general remarks on the notion of "determinism", I will discuss the precise meaning of chaos theory and the frequent misunderstandings concerning the implications of that theory. After reviewing the status of probabilistic reasoning in classical physics, I will also briefly discuss misunderstandings occurring in the context of quantum mechanics.

Approach to Equilibrium for the Phonon Boltzmann Equation

Communications in Mathematical Physics, 2008

We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a... more We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved quantities that diffuse to zero, and fast variables that are slaved to the slow ones. This slaving implies the Fourier law, which relates the induced currents to the gradients of the conserved quantities.

Random Walks in Space Time Mixing Environments

Journal of Statistical Physics, 2009

We prove that random walks in random environments, that are exponentially mixing in space and tim... more

In nite dimensional SRB-measures mp arc Preprint

Defense of a Modest Scientific Realism

Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories

Communications in Mathematical Physics, 1979

Correlation inequalities are used to show that the two component λ(φ2)2 model (with HD, D, HP, P ... more Correlation inequalities are used to show that the two component λ(φ2)2 model (with HD, D, HP, P boundary conditions) has a unique vacuum if the field does not develop a non-zero expectation value. It follows by a generalized Coleman theorem that in two space-time dimensions the vacuum is unique for all values of the coupling constant. In three space-time dimensions the vacuum is unique below the critical coupling constant. For then-componentP(|φ|2)2+μφ1 model, absence of continuous symmetry breaking, as μ goes to zero, is proven for all states which are translation invariant, satisfy the spectral condition, and are weak* limit points of finite volume states satisfyingN locτ and higher order estimates.

Renormalizing the renormalization group pathologies

Physics Reports-review Section of Physics Letters, 2001

We review the status of the “pathologies” of the Renormalization Group (RG) encountered when one ... more We review the status of the “pathologies” of the Renormalization Group (RG) encountered when one tries to define rigorously the RG transformation as a map between Hamiltonians. We explain their origin and clarify their status by relating them to the Griffiths’ singularities appearing in disordered systems; moreover, we suggest that the best way to avoid those pathologies is to use the contour representation rather than the spin representation for lattice spin models at low temperatures. Finally, we outline how to implement the RG in the contour representation.

On the derivation of Fourier's law for coupled anharmonic oscillators

We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three... more We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice and subjected to a stochastic forcing mimicking heat baths of temperatures T_1 and T_2 on the hyperplanes at x_1=0 and N. We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for N large, is approximately a local equlibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile.

First-order phase transitions in Potts and Ising systems

Physics Letters A, 1983

We use an equivalence between the q-state Potts model on a d-dimensional lattice for q = 2n, n = ... more We use an equivalence between the q-state Potts model on a d-dimensional lattice for q = 2n, n = 1, 2 … and a spin Ising model on n copies of the lattice with 2 n-spin ferromagnetic interactions to obtain some exact new results for both systems. In particular we prove the existence of first-order phase transitions for a class of ferromagnetic Ising systems obtained as perturbations of the large q Potts-Ising system and the non-vanishinng of the surface tension for the Potts model (with q = 2n) at all temperatures below their transition temperature.

The hierarchical random field Ising model

Journal of Statistical Physics, 1988

We introduce and solve explicitly a hierarchical approximation to the random field Ising model. T... more We introduce and solve explicitly a hierarchical approximation to the random field Ising model. This approximation is defined in terms of Peierls' contours. It exhibits a spontaneous magnetization ind>2 and illustrates some of the ideas used in the proof of that result for the real RFIM. Ind=2, there is no spontaneous magnetization, but a very slow decay of correlations. However, we argue that this latter property is an artifact of the approximation. For the real RFIM, we expect exponential decay of correlations for any value of the disorder.

Random surfaces in statistical mechanics

Inequalities between different classical spin models

Physics Letters A, 1976

We apply correlation inequalities to show that in a multicomponent spin model some quantities (e.... more

What is the meaning of the wave function?

I will review the constraints that the no hidden variable theorems put on the interpretations of ... more I will review the constraints that the no hidden variable theorems put on the interpretations of quantum mechanics and on the locality of the universe. I will also briefly explain how Bohm's theory gives a clear and objective meaning to the wave function and, at the same time, explains in the simplest possible way the most paradoxical aspects of quantum mechanics.