The spherical (Berlin–Kac) model driven by Ornstein–Uhlenbeck self-propulsion noise — the active spherical model — was solved statically by Ikeda (2022), who showed it has ordinary spherical-model (Ising-type) criticality for finite persistence time τp\tau_p and random-field-Ising-type criticality in the extreme-persistence limit τp→∞\tau_p\to\infty, with a persistence-dependent critical temperature. The paper explicitly leaves the model's out-of-equilibrium behavior open: no two-time correlation or response functions, no aging analysis after quenches, no assessment of fluctuation–dissipation-theorem violations or of an effective temperature, and no computation of the entropy production of the active steady state exist in the literature. The problem posed here is to construct this full out-of-equilibrium solution and its nonequilibrium thermodynamic characterization, verifying all equilibrium and static limits.

Background

The spherical model of Berlin and Kac is a solvable model of a ferromagnet: NN continuous spin variables xix_i interact through a quadratic Hamiltonian and are subject to the spherical constraint ∑i⟨xi2⟩=N\sum_i\langle x_i^2\rangle=N, enforced by a Lagrange multiplier μ\mu that plays a role analogous to the chemical potential of an ideal Bose gas. Because the model is Gaussian once μ\mu is fixed, it is exactly solvable in any spatial dimension dd and shows a condensation transition — a macroscopic occupation of the lowest normal mode — that is identified with the ferromagnetic phase transition. Ikeda (2022) considered a simplified version, the spherical Debye model (SDM), in which the normal-mode frequencies ωi\omega_i are distributed with the Debye density of states D(ω)=d ωD−d ωd−1D(\omega)=d\,\omega_D^{-d}\,\omega^{d-1} on 0≤ω≤ωD0\le\omega\le\omega_D (zero otherwise, normalized so that ∫0ωDD(ω) dω=1\int_0^{\omega_D}D(\omega)\,d\omega=1, with the Debye frequency set to ωD=1\omega_D=1) appropriate to phonon modes of a dd-dimensional lattice, and drove it with active matter physics: each mode obeys u˙i=−(μ+ωi2)ui+fi(t)\dot u_i=-(\mu+\omega_i^2)u_i+f_i(t), where the self-propulsion force fif_i is an Ornstein–Uhlenbeck process (a Gaussian noise with exponential time correlations ⟨fi(t)fj(t′)⟩=δij(T/τp)e−∣t−t′∣/τp\langle f_i(t)f_j(t')\rangle=\delta_{ij}(T/\tau_p)e^{-|t-t'|/\tau_p}) characterized by a persistence time τp\tau_p. For τp→0\tau_p\to0 the driving becomes thermal white noise; for finite τp\tau_p detailed balance is violated and the equipartition theorem fails, the steady second moments becoming ⟨ui2⟩=T/[(μ+ωi2)(1+τp(μ+ωi2))]\langle u_i^2\rangle=T/[(\mu+\omega_i^2)(1+\tau_p(\mu+\omega_i^2))]. From these moments alone Ikeda solved the statics: the critical temperature Tc(τp)T_c(\tau_p) rises with τp\tau_p — motility promotes ordering, echoing motility-induced phase separation (MIPS), the aggregation of self-propelled particles that motivated the model — and the critical exponents are those of the ordinary spherical model for finite τp\tau_p, while in the extreme-persistence limit τp→∞\tau_p\to\infty at fixed scaled temperature T~=T/τp\tilde T=T/\tau_p the frozen self-propulsion acts as a quenched random field and the exponents become those of the random-field spherical model, with lower and upper critical dimensions 4 and 6. The analysis is, however, purely static: it assumes a steady state with time-independent μ\mu and accesses only equal-time averages. For the equilibrium spherical model and its p-spin generalization (a spherical model with a pp-spin glass interaction), the out-of-equilibrium dynamics after a quench is a classical solved problem: two-time correlation and response functions exhibit aging (dependence on both times, not only their difference), and the fluctuation–dissipation theorem (FDT), which in equilibrium links the response RR to the correlation CC via R=(1/T)∂C/∂t′R=(1/T)\partial C/\partial t', is violated in a structured way that in glassy systems defines an effective temperature through the fluctuation–dissipation ratio. Whether such a program — full dynamical solution, fluctuation–dissipation ratio, effective temperature, and stochastic-thermodynamic entropy production (the rate at which the dynamics dissipates free energy, quantifying time-reversal symmetry breaking) — can be carried through for the active spherical model is the open question, since there the non-white driving breaks detailed balance already at the Gaussian level, in every dimension, and for arbitrary τp\tau_p.

Problem Statement

For the active spherical model — the spherical Debye model with normal modes obeying u˙i(t)=−(μ+ωi2)ui(t)+fi(t)\dot u_i(t)=-(\mu+\omega_i^2)u_i(t)+f_i(t), driven by Ornstein–Uhlenbeck self-propulsion forces τpf˙i=−fi+2Tηi\tau_p\dot f_i=-f_i+\sqrt{2T}\eta_i with persistence time τp\tau_p, and subject to the spherical constraint ∑i⟨ui2(t)⟩=N\sum_i\langle u_i^2(t)\rangle=N — derive the full out-of-equilibrium dynamical solution that the static analysis of Ikeda (2022) left open. Concretely: (1) compute the mode-resolved two-time correlation function Ci(t,t′)=⟨ui(t)ui(t′)⟩C_i(t,t')=\langle u_i(t)u_i(t')\rangle and the mode-resolved response function Ri(t,t′)=δ⟨ui(t)⟩/δhi(t′)R_i(t,t')=\delta\langle u_i(t)\rangle/\delta h_i(t') to a small conjugate field hih_i on mode ii (the full two-time functions of the original spin variables following by summation over modes), in the active steady state and after a quench (e.g. a sudden change of the noise strength TT at t=0t=0 from a stationary active preparation, with the self-propulsion forces continuing as Ornstein–Uhlenbeck processes), including the aging regime where the Lagrange multiplier μ\mu becomes time-dependent and must be determined self-consistently from the constraint; (2) quantify the violation of the fluctuation–dissipation theorem, e.g. through the mode-resolved fluctuation–dissipation ratio Xi(t,t′)=Ri(t,t′)/(∂Ci(t,t′)/∂t′)X_i(t,t')=R_i(t,t')/(\partial C_i(t,t')/\partial t') relative to its equilibrium value 1/T1/T, and determine whether an effective temperature (a temperature-like parameter inferred from the long-time fluctuation–dissipation ratio or from the slow-mode response) can be consistently defined for the active steady state, or whether it is observable-dependent; (3) compute the mean entropy production rate of the active steady state, quantifying time-reversal symmetry breaking of the driven dynamics; and (4) treat the extreme-persistence limit τp→∞\tau_p\to\infty at fixed scaled temperature T~=T/τp\tilde T=T/\tau_p as a distinguished case, where the frozen driving acts as a quenched random field. The solution must reproduce all known limits: τp→0\tau_p\to0 recovering the equilibrium spherical-model dynamics, equal-time quantities reproducing the known steady-state second moments, and the static transition temperatures Tc(τp)T_c(\tau_p) in both the finite- and infinite-persistence regimes.

References

  1. Harukuni Ikeda, 'Active Spherical Model', arXiv:2208.10200 (2022), DOI: 10.48550/arXiv.2208.10200, https://arxiv.org/abs/2208.10200
  2. Harukuni Ikeda, 'Correlated noise and critical dimensions', Phys. Rev. E 108, 064119 (2023), DOI: 10.1103/PhysRevE.108.064119, arXiv:2302.13666, https://arxiv.org/abs/2302.13666
  3. Harukuni Ikeda and Yuta Kuroda, 'Continuous symmetry breaking of low-dimensional systems driven by inhomogeneous oscillatory driving forces', Phys. Rev. E 110, 024140 (2024), DOI: 10.1103/PhysRevE.110.024140, arXiv:2304.14235, https://arxiv.org/abs/2304.14235
  4. Harukuni Ikeda, 'Harmonic chain far from equilibrium: Single-file diffusion, long-range order, and hyperuniformity', SciPost Phys. 17, 103 (2024), DOI: 10.21468/SciPostPhys.17.4.103, arXiv:2309.03155, https://arxiv.org/abs/2309.03155
  5. Harukuni Ikeda, 'Interaction-free ergodicity-breaking driven by temporally hyperuniform noise', Phys. Rev. E 112, 054136 (2025), DOI: 10.1103/g921-ppyv, arXiv:2503.06590, https://arxiv.org/abs/2503.06590
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