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We consider a model in which positive and negative particles with equaldensities diffuse in an asymmetric, CP invariant way on a ring. The positiveparticles hop clockwise, the negative counter-clockwise and oppositely-chargedadjacent particles may swap positions. The model depends on two parameters.Analytic calculations using quadratic algebras, inhomogeneous solutions of themean-field equations and Monte-Carlo simulations suggest that the model hasthree phases. A pure phase in which one has three pinned blocks of onlypositive, negative particles and vacancies and in which translationalinvariance is broken. A mixed phase in which the current has a lineardependence on one parameter but is independent of the other one and of thedensity of the charged particles. In this phase one has a bump and a fluid. Thebump (condensate) contains positive and negative particles only, the fluidcontains charged particles and vacancies uniformly distributed. The mixed phaseis separated from the disordered phase by a second-order phase-transition whichhas many properties of the Bose-Einstein phase-transition observed inequilibrium. Various critical exponents are found.Comment: 58 pages, published versio

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring: I. The Neutral System