Abstract
We consider a d-dimensional gas in canonical equilibrium under pairwise screened Coulomb repulsion and external confinement, and subject to a volume constraint (hard walls). We show that its excess free energy displays a third-order singularity separating the pushed and pulled phases, irrespective of range of the pairwise interaction, dimension and details of the confining potential. The explicit expression of the excess free energy is universal and interpolates between the Coulomb (long-range) and the delta (zero-range) interaction. The order parameter of the transition—the electrostatic pressure generated by the surface excess charge—is determined by invoking a fundamental energy conservation argument.
Highlights
The understanding of when and how phase transitions occur—i.e. instances whereby certain properties of a medium change, often abruptly, as a result of the change of some external condition, such as temperature, pressure, or others—is one of the most striking successes of classical statistical mechanics
When N → ∞, the equilibrium configuration can be characterized by the density of the gas, which, under general assumptions [33, 8], is the minimizer of the mean-field free energy functional at zero temperature
The formulation of the problem in such general terms allows us to identify the previously elusive order parameter—a quantity that vanishes in one phase but is nonzero in the other—of this third-order transition: the ‘electrostatic’ pressure generated by the surface excess charge
Summary
The understanding of when and how phase transitions occur—i.e. instances whereby certain properties of a medium change, often abruptly, as a result of the change of some external condition, such as temperature, pressure, or others—is one of the most striking successes of classical statistical mechanics. When N → ∞, the equilibrium configuration can be characterized by the density of the gas, which, under general assumptions [33, 8], is the minimizer of the mean-field free energy functional at zero temperature. The formulation of the problem in such general terms allows us to identify the previously elusive order parameter—a quantity that vanishes in one phase (pulled) but is nonzero in the other (pushed)—of this third-order transition: the ‘electrostatic’ pressure generated by the surface excess charge. The first task is to compute the constrained equilibrium density R(x) corresponding to the interaction kernel Φd, which is the minimizer of the quadratic functional (2)
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