Response theory and phase transitions for the thermodynamic limit of interacting identical systems

Abstract

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers-Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker-Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai-Zwanzig model and of the Bonilla-Casado-Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.

Keywords: Bonilla–Casado–Morilla model; Desai–Zwanzig model; Kramers–Kronig relations; order–disorder transitions; sum rules; thermodynamic limit.

Figure 1.

Results of numerical simulations of equation (6.1) with α = 1. Heat maps of the order parameter 〈x〉₀ (panel (a)); of the re-scaled variance (θ/σ²)〈z²〉₀ (panel (b)); and of the rescaledcorrelation time $\hat{\tau }=\theta \times \tau$ (panel (c)). The dotted red line shows the transition line, see [1]. See text for details. (Online version in colour.)

Figure 2.

A horizontal (left) and a vertical (right) section of the heat maps shown in figure 1a–c. (a) From top to bottom: order parameter, rescaled variance and rescaled integrated autocorrelated time as a function of the strength of the noise. Here θ ≈ 0.4. (b) From top to bottom: order parameter, rescaled variance and rescaled integrated autocorrelated time as a function of the strength of the coupling. Here σ ≈ 0.78. (Online version in colour.)

Figure 3.

Results of numerical simulations of equation (6.11) with α = 2. Heat maps of the amplitude A₁ of the oscillations of the variable x (panel (a)), of the amplitude A₂ of the oscillationsof the re-scaled variance θ/σ²〈z²〉 (panel (b)), and of the time mean value of θ/σ²〈z²〉 (panel (c)). The red dotted line represents the transition line given by equation (6.21); see [89]. See text for details. (Online version in colour.)

Figure 4.

Horizontal (left) and vertical (right) sections of the heat plots 3a–c. (a) From top to bottom: A₁(α = 2, σ, θ = 2), A₂(α = 2, σ, θ = 2), and B₂(α = 2, σ, θ = 2). (b) From top to bottom: A₁(α = 2, σ = 1.6, θ), A₂(α = 2, σ = 1.6, θ), and B₂(α = 2, σ = 1.6, θ). (Online version in colour.)