Below its critical temperature, the spherical lattice gas of Gersch and Berlin has a translation-invariant Gibbs state that is a mixed phase: a uniform mixture over a continuously degenerate family of pure macrostates, each describing a single diffuse droplet of condensed matter (a cosine-mode condensate). Patrick's exact solution of the droplet shapes (J. Stat. Phys. 142, 1085-1104, 2011) left the dynamical counterpart open: whether transitions between these pure macrostates occur on experimentally long or short timescales was never evaluated, although the author noted that the Langevin-dynamics techniques developed for the mean spherical model should allow the calculation and that static large-deviation results indicate a free-energetic cost on the scale nd−2n^{d-2}. The problem is to specify a stochastic dynamics faithful to the conventional (exactly constrained) spherical lattice gas and to compute the interstate transition times for d≥3d\ge3 below TcT_c: their mechanism (activated barrier crossing versus diffusion along the degenerate condensate manifold), their scaling with system size, and the resulting answer to whether an experiment would observe a pure droplet macrostate or the mixed phase.

Background

The Berlin-Kac spherical model replaces the discrete spins of the Ising model by real variables xj∈Rx_j\in\mathbb{R} on the sites of a hypercubic lattice, coupled by a nearest-neighbour ferromagnetic Hamiltonian and subject to the spherical constraint ∑jxj2=N\sum_j x_j^2=N (with NN the number of sites); the model is exactly solvable by Fourier diagonalization and has a phase transition only for d≥3d\ge3. The lattice-gas version of the model (Gersch and Berlin, 1962) adds a density constraint 1N∑jxj=ρ\frac{1}{N}\sum_j x_j=\rho, so that the variables behave like continuous occupation numbers; below a critical temperature the model separates into a dilute gas phase and a condensed liquid phase, with an exactly known equation of state. Gersch and Berlin also found that the two-point covariances of the translation-invariant Gibbs state do not decay with distance: in modern language the equilibrium state is a mixed phase (a convex combination of extremal components) rather than a pure phase, in which covariances would decay (the clustering property). The pure components describe localized droplets, but their shape was not extracted until Patrick's 2011 analysis, which used generalized quasi-averages: the Bogolubov quasi-average prescription (switch on a weak symmetry-breaking field, take the thermodynamic limit, then let the field magnitude tend to zero) in the generalized form of Angelescu and Zagrebnov, where the field couples to a cosine mode with magnitude ϵn−δ\epsilon n^{-\delta}; for decay rates 1<δ<d−11<\delta<d-1 the field selects a pure state without deforming it, while slower decay (δ≤1\delta\le1) deforms the droplet and faster decay (δ≥d−1\delta\ge d-1) fails to select a pure state at all. The result is that a pure macrostate is a single-cosine-mode condensate with expected density profile ρ+2(1−ρ2)(1−βc/β)cos⁡(2πj1/n−α)\rho+\sqrt{2(1-\rho^2)(1-\beta_c/\beta)}\cos(2\pi j_1/n-\alpha): a droplet whose boundary is diffuse on every scale, in contrast with the sharp Wulff-shape droplets (equilibrium crystal shapes obtained by minimizing interfacial free energy) proven for discrete Ising-type lattice gases. The pure states form a continuously degenerate family: at zero temperature they are ρ+2(1−ρ2)∑νrνcos⁡(2π(jν−1)/n−αν)\rho+\sqrt{2(1-\rho^2)}\sum_\nu r_\nu\cos(2\pi(j_\nu-1)/n-\alpha_\nu) with ∑νrν2=1\sum_\nu r_\nu^2=1, so the condensate direction is free to move on a sphere inside the 2d2d-dimensional eigenspace of the second-largest eigenvalue λs\lambda_s of the periodic interaction matrix; the natural Gibbs state is the uniform mixture over this family, with the droplet location uniformly distributed. Which of these does a physical system actually exhibit? Patrick's discussion observes that the answer is dynamical: if the typical transition time between pure states far exceeds the experimental observation time, measurements yield one pure state; if it is much shorter, they yield the mixed state. The dynamical technology exists for the mean spherical model, the variant in which the spherical constraint is imposed only on average through a time-dependent Lagrange multiplier: its Langevin dynamics decomposes into independent Ornstein-Uhlenbeck processes (relaxing Gaussian Fourier modes) coupled by a Volterra integral equation for the multiplier (Godreche and Luck, 2000). Patrick states that the interstate transition times 'can be calculated using the technique proposed there', that he had 'not attempted yet those calculations', and that static large-deviation results (probabilities of atypical magnetization values decaying like exp⁡(−nd−2)\exp(-n^{d-2}), governed by a rate function; see also Patrick's 2012 rate-function analysis) indicate a free-energetic cost on the scale nd−2n^{d-2} for a transition, making transition times plausibly much longer than measurement times. Two caveats structure the open problem. First, the conventional model (exact constraint) and the mean spherical model are known to become dynamically inequivalent beyond a crossover time t∗∼V2/dt^*\sim V^{2/d} (Fusco and Zannetti, 2002), so dynamical results for the mean model do not automatically transfer. Second, because the pure-state family is continuously degenerate (the interaction energy depends only on the total condensate norm, not its direction within the λs\lambda_s eigenspace), it is not a priori clear whether interstate transitions are activated barrier crossings or barrier-free diffusion of the condensate direction along the degenerate manifold; establishing the mechanism and the resulting timescale is part of the question.

Problem Statement

For the conventional spherical lattice gas (real occupation variables xjx_j on the periodic cube Vn⊂ZdV_n\subset\mathbb{Z}^d of side length nn, so that N=ndN=n^d is the number of sites, with d≥3d\ge3; exact density constraint 1N∑jxj=ρ\frac{1}{N}\sum_j x_j=\rho and spherical constraint ∑jxj2=N\sum_j x_j^2=N; nearest-neighbour ferromagnetic interaction) at inverse temperature β>βc(ρ)\beta>\beta_c(\rho), the critical value separating the condensed phase from the uniform gas: (1) specify a stochastic dynamics faithful to this model, meaning a dynamics that preserves both constraints and has the spherical-lattice-gas Gibbs measure as a stationary measure; admissible choices include a reversible Langevin diffusion on the constraint sphere, or a justified adaptation of the mean-spherical (kinetic) dynamics of Godreche and Luck, with the conventional-versus-mean distinction addressed explicitly rather than assumed away. (2) For that dynamics, calculate the typical transition times between distinct pure droplet macrostates, realized at finite nn as configurations whose condensate (the projection of {xj}\{x_j\} onto the 2d2d-dimensional eigenspace of λs\lambda_s, the second-largest eigenvalue of the nearest-neighbour interaction matrix) lies in a neighbourhood of a given direction: determine the mechanism, that is, activated crossing of a free-energy barrier (whose existence and scale must be established; the source suggests a barrier on the scale nd−2n^{d-2}, that is, transition probabilities exponentially small in nd−2n^{d-2}) versus barrier-free diffusion of the condensate direction along the degenerate eigensphere, and the resulting scaling of the transition time with nn (for example exp⁡(c nd−2)\exp(c\,n^{d-2}) versus polynomial in nn), stating how the inter-state neighbourhoods are chosen and addressing the dependence of the claimed scaling on that choice. (3) Answer the physical question posed in the source: in the thermodynamic limit, is the transition time between pure macrostates much greater than any fixed observation time, so that an experiment observes a single pure droplet macrostate, or does the system explore the family of pure states, and hence effectively the mixed state, on bounded timescales? Since the source poses this question at the level of the system itself, the analysis of (1)-(2) may be carried out for a single, explicitly specified faithful dynamics chosen from (1); if the resulting mechanism or scaling is claimed to hold for the model rather than for that particular dynamics, the dependence on the choice made in (1) must be identified and the system-level conclusion justified accordingly. The companion static question raised in the same sentence of the source (whether the droplet shapes of the mean spherical and the conventional model coincide) may be addressed en route but is not required for completion.

References

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