Variational Mechanics of Moral Hysteresis: A McKean–Vlasov Framework for Normative Collapse and Recovery

Variational Mechanics of Moral Hysteresis: A McKean–Vlasov Framework for Normative Collapse and Recovery | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Variational Mechanics of Moral Hysteresis: A McKean–Vlasov Framework for Normative Collapse and Recovery regio marcos pinto abreu filho This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8223366/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We propose a mean-field model in which the collapse and recovery of social norms appear as hysteretic transitions in a population of interacting agents. Each agent carries a continuous moral state xₜ ∈ ℝ, interpreted as a coarse-grained propensity to choose norm-compliant rather than norm-violating actions in a given context. The dynamics follow an interacting diffusion dxₜ = −∂ₓV(xₜ, mₜ; λ) dt + √(2D) dWₜ, where V(x, m; λ) is an effective potential, mₜ is the population mean moral state, λ is a control parameter (e.g. antisocial reward or perceived corruption level), D is a noise amplitude, and Wₜ is a Wiener process. Social influence enters via a mean-field coupling of strength κ and a prosocial field h. In the large-population limit, the empirical distribution of moral states converges to a solution of a nonlinear Fokker–Planck (McKean–Vlasov) equation. For a double-well potential with mean-field coupling, the stationary solutions exhibit monostable and bistable regimes as (λ, κ, D, h) vary. Under quasi-static driving of λ from low to high values and back, starting from high- and low-norm initial conditions, the system traces a hysteresis loop in the (λ, m*) plane, where m*(λ) is the quasi-stationary mean at fixed λ. We define collapse and recovery thresholds λ_coll and λ_rec, the hysteresis width Δλ = λ_coll − λ_rec, and the loop area A_hyst and estimate them numerically from finite-population simulations. We then introduce a McKean–Vlasov free-energy functional to interpret these phenomena in variational terms, distinguishing binodal lines (thermodynamic coexistence of high- and low-norm phases) from spinodal lines (loss of dynamical stability of a given phase), and we sketch Kramers-type estimates for escape times between basins. In the parameter regimes explored, stronger social coupling widens the hysteresis loop, intermediate noise facilitates recovery from low-norm regimes, and variance, autocorrelation and recovery time of the mean moral state increase near λ_coll, consistent with critical slowing down. We conclude by discussing how this formalism could be connected, at least qualitatively, to behavioural experiments and survey indicators of norm stability and change. Physical sciences/Mathematics and computing Physical sciences/Physics Figures Figure 1 1. Introduction Episodes of breakdown and partial recovery of social norms are widely documented in political science, moral psychology and institutional economics. Periods of relative stability, in which prosocial restraint and rule compliance are common, can be interrupted by transitions to low-norm regimes with increased tolerance for corruption, opportunism or violence. Recovery to previous high-norm states, when it occurs at all, is often slow and asymmetric: simply reversing the external conditions that preceded collapse does not automatically restore earlier norms. From a dynamical-systems perspective, this behaviour suggests multiple attractors and metastability. Societies may spend long periods near a high-norm equilibrium, then cross a threshold beyond which a low-norm equilibrium becomes dominant, and later require substantially improved conditions to return to high norms. Similar mechanisms underlie first-order phase transitions and hysteresis in statistical mechanics, where systems exhibit coexisting phases, metastable states and path dependence under slow parameter changes. In statistical physics, the Curie–Weiss model of ferromagnetism provides a canonical example of such behaviour. As an external field is slowly increased and decreased, magnetisation follows different branches depending on history, due to the coexistence of locally stable states separated by energy barriers. Socio-physics and opinion-dynamics models have adopted related mechanisms to describe cascades, contagion and collective shifts in attitude, often using mean-field couplings and nonlinear drift terms. Empirical work on norms and cooperation emphasises that moral behaviour is strongly context-dependent. Payoff structures, enforcement, institutional signals and the observed behaviour of peers shape willingness to comply with norms and punish violations. Existing mathematical models capture some aspects of this, but few connect moral interpretation explicitly to the hysteretic and variational structure of interacting diffusions and McKean–Vlasov limits. The aim of this paper is to formulate a simple McKean–Vlasov model of “moral hysteresis” and to analyse its qualitative behaviour using tools from variational mechanics and mean-field statistical physics. At the microscopic level, each agent is described by a scalar moral state xₜ evolving in an effective potential V(x, m; λ) under noise and mean-field coupling. At the macroscopic level, we study the induced nonlinear Fokker–Planck equation and its stationary solutions, introduce a free-energy functional with high- and low-norm minima, and investigate collapse and recovery under quasi-static driving of the control parameter. We illustrate how a double-well potential with mean-field coupling can generate a hysteresis loop in the (λ, m) plane, how the loop width and area depend on social coupling and noise, and how early-warning indicators behave near collapse. The model is deliberately stylised; the focus is on mechanisms rather than on quantitative calibration. 2. Model: interacting moral diffusions and mean-field limit 2.1 Agent-level dynamics We consider a population of N agents indexed by i = 1, …, N. Each agent has a scalar moral state xᵢ(t) ∈ ℝ at time t ≥ 0, interpreted as a coarse-grained propensity to act in accordance with norms in a given context. Positive values correspond to strong prosocial tendencies; negative values correspond to a higher propensity to choose norm-violating actions. The empirical mean moral state is m_N(t) = (1/N) Σᵢ xᵢ(t). The dynamics of each agent are given by the stochastic differential equation dxᵢ(t) = −∂ₓ V(xᵢ(t), m_N(t); λ(t)) dt + √(2D) dWᵢ(t), where D > 0 is a noise amplitude, {Wᵢ} are independent standard Wiener processes, and V(x, m; λ) is an effective potential of the form V(x, m; λ) = (a/4) x⁴ − (b(λ)/2) x² − κ m x − h x. Here a > 0 is fixed, b(λ) is a smooth function of the control parameter λ, κ ≥ 0 is the social coupling strength, and h is a constant prosocial field. This potential combines three contributions: a quartic term (a/4) x⁴ ensuring stability at large |x|; a quadratic term −(b(λ)/2) x² whose sign and magnitude control whether the potential is monostable or double-well, and how sensitive it is to λ; a linear term −(κ m + h) x representing social influence from the mean moral state and an external prosocial bias. In the simplest implementation we take b(λ) = b₀ − b₁ λ with b₀, b₁ > 0, so that increasing λ effectively favours low-norm states by shallowing the high-norm well and deepening the low-norm well. Figure 1 (conceptual) shows a schematic example of V(x, m; λ) at m = 0 for three values of λ, illustrating the emergence and disappearance of distinct wells as the control parameter changes. 2.2 McKean–Vlasov limit In the limit N → ∞ with appropriate assumptions on initial conditions, the empirical measure of (xᵢ(t)) converges to a deterministic probability measure ρₜ on ℝ solving a nonlinear Fokker–Planck (McKean–Vlasov) equation. Formally, if ρₜ(dx) ≈ (1/N) Σᵢ δ_{xᵢ(t)}(dx), then ρₜ satisfies ∂ₜ ρₜ(x) = ∂ₓ [ ∂ₓ V(x, mₜ; λ(t)) ρₜ(x) ] + D ∂ₓₓ ρₜ(x), where mₜ = ∫ℝ x ρₜ(x) dx is the mean moral state induced by ρₜ. This is a McKean–Vlasov or mean-field interacting diffusion, in which the drift depends on ρₜ via mₜ. For fixed λ and parameters (κ, D, h), stationary solutions of this equation are probability densities ρ*(x; λ) satisfying 0 = ∂ₓ [ ∂ₓ V(x, m*; λ) ρ*(x; λ) ] + D ∂ₓₓ ρ*(x; λ), together with the self-consistency condition m*(λ) = ∫ℝ x ρ*(x; λ) dx. For a double-well potential, there can be one or several stationary solutions satisfying these equations, leading to monostable or bistable regimes. 2.3 Moral interpretation of parameters The abstract variables of the model admit a concrete interpretation in terms of norms and institutions: xᵢ(t) is a latent moral state that can be related to the log-odds of choosing a prosocial action in a repeated game or allocation task. mₜ is the average of these log-odds and serves as an order parameter summarising the moral climate. λ acts as a control parameter representing effective antisocial reward or perceived corruption level, combining payoff skew and enforcement. κ quantifies how strongly individuals are pulled towards the current population mean, capturing social influence, conformity or contagion. h is a prosocial field reflecting institutional messaging, reputational incentives or explicit normative cues. D captures idiosyncratic fluctuations in moral decision-making and unmodelled heterogeneity. The model is stylised, but this mapping indicates how its parameters could be related to empirical quantities. 3. Free-energy structure and variational interpretation 3.1 McKean–Vlasov free energy For fixed λ, a stationary solution of the McKean–Vlasov equation can be written, at least formally, as a Gibbs measure ρ*(x; λ) ∝ exp[ − (1/D) V(x, m*; λ) ], with m* defined by the self-consistency condition above. More generally, one can define a McKean–Vlasov free-energy functional on the space of probability measures ℱ_λ[ρ]: ℱ_λ[ρ] = ∫ℝ V(x, m[ρ]; λ) ρ(dx) + D ∫ℝ ρ(x) log ρ(x) dx, where m[ρ] = ∫ x ρ(dx). Under appropriate conditions, stationary solutions correspond to critical points of ℱ_λ, and dynamically stable solutions are local minima of ℱ_λ. In our context, high-norm and low-norm regimes appear as distinct local minima ρ_high and ρ_low with different mean values m _{high} and m _{low}. 3.2 Binodal and spinodal lines As λ varies, the structure of the free-energy landscape ρ ↦ ℱ_λ[ρ] changes. There may be ranges of λ where: a single minimum exists (monostable regime); two minima exist with comparable depth (bistable regime with potential coexistence); one of the minima is shallow and close to losing stability (near a spinodal point). It is therefore useful to distinguish: a binodal interval [λ_b,low, λ_b,high] where both high- and low-norm minima exist and have comparable free energy; spinodal points λ_s,high and λ_s,low at which one of these minima ceases to exist or loses dynamical stability. In classical thermodynamics, binodal lines characterise coexistence of phases, while spinodal lines mark the onset of absolute instability. In the present interpretation, binodal intervals mark parameter ranges in which both high- and low-norm regimes could, in principle, persist, while spinodal points indicate thresholds at which one regime can no longer be sustained against small perturbations. Under slow driving of λ, the system can remain trapped in a metastable minimum beyond the binodal interval, up to the corresponding spinodal point. This leads to hysteresis: the transition from high norms to low norms on the upward sweep occurs near λ_s,high, whereas the recovery on the downward sweep occurs near λ_s,low. 3.3 Escape times and critical slowing down In finite populations with noise D > 0, transitions between metastable minima occur via rare fluctuations. Large-deviation theory and Kramers-type arguments suggest that the mean escape time from a metastable basin scales roughly as τ_esc ∼ exp(Δℱ / D), where Δℱ is the free-energy barrier between the metastable minimum and the relevant saddle. In this model, this implies that a population can remain trapped in a low-norm regime long after a high-norm configuration becomes globally favourable, if the free-energy barrier is sufficiently high and D is small. Near spinodal points, the curvature of ℱ_λ at the relevant minimum decreases and the associated relaxation time grows. A standard Landau-type approximation leads to an effective equation for the mean moral state mₜ of the form dmₜ/dt = −∂ₘ Φ(mₜ; λ), with an effective potential Φ(m; λ) whose curvature at a stable fixed point m*(λ) scales as ∂²Φ/∂m² ∝ |λ − λ_c| near a saddle-node bifurcation at λ_c. Linearising around m* gives a relaxation time τ(λ) ∝ |λ − λ_c|⁻¹, capturing critical slowing down. In simulations, we observe increasing variance, lag-1 autocorrelation and recovery times for the mean moral state as λ approaches the estimated collapse threshold, consistent with this qualitative picture. 4. Numerical methodology We study the model numerically using finite-population simulations of the agent-based SDE system. All simulations use the Euler–Maruyama scheme with a fixed time step Δt and are implemented in Python. 4.1 Baseline parameters Baseline parameters used in simulations (illustrative values; to be replaced by final calibrated numbers): N = 10⁴ (number of agents) Δt = 0.01 (time step) D = 0.05 (noise amplitude) a = 1.0 (quartic coefficient in V) b₀ = 1.0 (baseline quadratic coefficient) b₁ = 0.7 (sensitivity of b(λ) to λ) κ = 0.6 (social coupling strength, reference case) h = 0.0 (prosocial field, zero in reference case) λ_min = −0.5 (lower bound of quasi-static sweep) λ_max = 1.5 (upper bound of quasi-static sweep) 4.2 Euler–Maruyama discretisation For a given parameter set (N, Δt, D, κ, h) and fixed λ, the Euler–Maruyama update for agent i is xᵢⁿ⁺¹ = xᵢⁿ − ∂ₓ V(xᵢⁿ, mⁿ; λ) Δt + √(2D Δt) ξᵢⁿ, where mⁿ = (1/N) Σᵢ xᵢⁿ and ξᵢⁿ ∼ Normal(0, 1) are i.i.d. standard normal increments. The potential V is given by V(x, m; λ) = (a/4) x⁴ − (b(λ)/2) x² − κ m x − h x with b(λ) = b₀ − b₁ λ unless stated otherwise. 4.3 Quasi-static driving of the control parameter To probe hysteresis, we apply a quasi-static protocol to the control parameter λ. We choose bounds λ_min < λ_max and define a grid λ₁, …, λ_L of L equally spaced values in [λ_min, λ_max]. For the ascending (collapse) branch: Initialise all agents in a high-norm state at λ = λ_min (for example, a narrow Gaussian around a positive x₀). For each λ_k in increasing order: Integrate the dynamics for n_relax steps to discard transients. Then integrate for n_avg steps and compute the time-averaged mean moral state m⁺(λ_k). Use the final configuration at λ_k as initial condition for λ_{k+1}. For the descending (recovery) branch: Initialise all agents in a low-norm state at λ = λ_max (for example, around a negative x₀). Sweep λ downwards over the same grid, repeating the relax-and-average protocol to obtain m⁻(λ_k). Reverse the order of {m⁻(λ_k)} to align it with {λ_k}. 4.4 Hysteresis metrics Given the ascending and descending curves (λ_k, m⁺(λ_k)) and (λ_k, m⁻(λ_k)), we define: Collapse threshold λ_coll: the smallest λ_k on the ascending branch where m⁺(λ_k) crosses a chosen reference level (by default zero) from above. Recovery threshold λ_rec: the largest λ_k on the descending branch where m⁻(λ_k) crosses the same reference level from below. Hysteresis width: Δλ = λ_coll − λ_rec (when both thresholds are defined). Hysteresis area A_hyst, estimated by the trapezoidal rule, e.g.: A_hyst ≈ Σ_{k=1}^{L−1} ½ [ (m⁺(λ_k) − m⁻(λ_k)) + (m⁺(λ_{k+1}) − m⁻(λ_{k+1})) ] (λ_{k+1} − λ_k). In practice we repeat the procedure for R independent realisations (random seeds) per parameter cell and report ensemble means and variability. 4.5 Early-warning indicators To study early-warning indicators near collapse, we select a set of λ-values approaching the estimated λ_coll from below on the ascending branch. For each such λ we: Run long simulations at fixed λ, initialising in a high-norm state. Discard an initial transient. Record the mean moral state m(t) over a long time window. From these time series we estimate: Stationary variance Var*(λ) of m(t). Lag-1 autocorrelation AC₁(λ) of m(t). Recovery time τ_rec(λ) after small perturbations away from the quasi-stationary mean. We then analyse the dependence of these indicators on distance d(λ) = λ_coll − λ. 5. Results We summarise here the main qualitative patterns obtained in simulations of the model. Numerical values shown in figures and tables in this narrative are illustrative and should be replaced by estimates from full parameter sweeps. 5.1 Hysteresis in the (λ, m) plane For fixed (κ, D, h), the quasi-static protocol yields ascending and descending branches λ ↦ m⁺(λ) and λ ↦ m⁻(λ). A representative hysteresis loop in the (λ, m) plane shows that: starting from high norms, m⁺(λ) remains near the high-norm branch until λ reaches a collapse threshold λ_coll, at which point m⁺(λ) drops towards the low-norm branch; starting from low norms, m⁻(λ) remains low until λ is decreased to a recovery threshold λ_rec < λ_coll, at which point it returns to high norms. The width Δλ = λ_coll − λ_rec is positive and the loop area A_hyst is non-zero, indicating a genuine hysteretic regime. These quantities are used as summary metrics of path dependence. 5.2 Effect of social coupling To examine the role of social influence, we vary the coupling strength κ while keeping (D, h) and all numerical settings fixed. For each κ in a grid we compute hysteresis loops and extract λ_coll, λ_rec, Δλ and A_hyst. Over the range explored: the hysteresis width Δλ tends to increase with κ; the loop area A_hyst also increases with κ. For weak coupling (κ close to zero), the ascending and descending branches are close together and the hysteresis loop is narrow. As κ grows, the gap between branches widens and the collapse and recovery thresholds separate further, making path dependence more pronounced. Illustrative numbers (to be replaced): κ = 0.0: Δλ ≈ 0.05, A_hyst ≈ 0.010 κ = 0.2: Δλ ≈ 0.12, A_hyst ≈ 0.035 κ = 0.4: Δλ ≈ 0.20, A_hyst ≈ 0.070 κ = 0.6: Δλ ≈ 0.28, A_hyst ≈ 0.110 κ = 0.8: Δλ ≈ 0.35, A_hyst ≈ 0.150 5.3 Effect of noise We next vary the noise amplitude D while holding (κ, h) fixed. For each D we estimate hysteresis loops and recovery properties. The qualitative pattern is typically non-monotonic: For very small D, the system behaves almost deterministically: once it enters a given basin, it remains there for the duration of the simulation unless λ is pushed beyond a spinodal point. Recovery from a low-norm state is rare unless λ is reduced far below the binodal interval. As D increases to moderate levels, escape from low-norm basins becomes more frequent, and recovery can occur at higher values of λ. At very high noise, the distinction between high- and low-norm states becomes blurred, and the hysteresis loop becomes less well defined. Illustrative example for recovery probability (within a fixed time window) and conditional mean recovery time τ_rec(D): D = 0.01: recovery probability ≈ 0.10, τ_rec ≈ 4500 D = 0.03: recovery probability ≈ 0.45, τ_rec ≈ 2200 D = 0.05: recovery probability ≈ 0.75, τ_rec ≈ 1400 D = 0.08: recovery probability ≈ 0.65, τ_rec ≈ 1600 D = 0.12: recovery probability ≈ 0.40, τ_rec ≈ 2600 5.4 Early-warning indicators Finally, we examine early-warning indicators near the collapse threshold. Fixing (κ, D, h), we select a grid of λ-values approaching the estimated λ_coll from below and compute variance, lag-1 autocorrelation and recovery time of m(t) as described in the methods. Typical patterns: Stationary variance Var*(m) increases as λ approaches λ_coll from below. Lag-1 autocorrelation AC₁(m) also increases, approaching values close to 1 near collapse. Recovery times τ_rec become longer as λ approaches λ_coll, reflecting critical slowing down. If we define d(λ) = λ_coll − λ, we typically see Var*(m) and AC₁(m) as increasing functions of 1/d(λ) or decreasing functions of d(λ). These patterns are qualitatively consistent with the theory of saddle-node bifurcations and early-warning signals in complex systems. 6. Discussion We have introduced a McKean–Vlasov model in which social norms appear as metastable states in an effective moral potential, and in which the transition between high-norm and low-norm regimes exhibits hysteresis under quasi-static driving of an antisocial reward parameter. By connecting the model to a free-energy functional, we distinguish thermodynamic coexistence (binodal) from dynamical instability (spinodal) and outline how escape times and critical slowing down arise from the structure of the free-energy landscape. The model is coarse-grained: moral state is one-dimensional, interactions are mean-field, and environmental complexity is compressed into a single scalar parameter λ. We do not claim empirical realism at the level of detailed prediction; the model should be viewed as a theoretical testbed for mechanisms. Within this stylised setting, the simulations suggest several robust qualitative messages: there is a genuine hysteretic regime, with distinct collapse and recovery thresholds; stronger social coupling κ widens the hysteresis loop and increases the loop area; moderate noise D can facilitate recovery from low norms, whereas very low and very high noise can impede it; early-warning indicators such as variance, autocorrelation and recovery time behave as expected near a saddle-node bifurcation. Although the present analysis is theoretical, the structure of the model suggests ways to link it to empirical data. In repeated-game experiments, λ could be implemented as a payoff ratio favouring defection over cooperation, κ as the strength of feedback about others’ choices, and h as explicit normative messaging. By slowly varying λ and measuring population-averaged behaviour, one could estimate collapse and recovery thresholds and compare them to model predictions. In observational settings, macro-level indicators of trust, tolerance for corruption, or willingness to punish norm violations might serve as proxies for m*(λ), while institutional changes or economic shocks would play the role of shifts in λ. Several extensions are worth pursuing. First, it would be natural to move beyond a single scalar moral variable to a multi-dimensional state capturing different moral dimensions (e.g. harm, fairness, authority). Second, introducing network structure and heterogeneity in interactions would permit the study of localised moral collapse, cluster formation and differential resilience across communities. Third, a fuller mathematical treatment of the McKean–Vlasov free-energy landscape and its large-deviation properties, building on existing results for interacting diffusions, would clarify the relationship between microscopic parameters and macroscopic hysteresis observables. Declarations Author Contribution Author contributions.R.M.A.F. designed the model, carried out the analytical derivations, implemented and ran the simulations, processed and visualised the data, and wrote and edited the manuscript. Acknowledgements [Acknowledgements to be completed.] References C. Castellano, S. Fortunato and V. Loreto, “Statistical physics of social dynamics,” Reviews of Modern Physics, 81, 591–646, 2009. M. Kochmański, T. Paszkiewicz and S. Wolski, “Curie–Weiss magnet — a simple model of phase transition,” European Journal of Physics, 34(6), 1555, 2013. M. Scheffer, J. Bascompte, W. A. Brock et al., “Early-warning signals for critical transitions,” Nature, 461(7260), 53–59, 2009. C. Bicchieri, The Grammar of Society: The Nature and Dynamics of Social Norms . Cambridge University Press, 2006. E. Fehr and S. Gächter, “Cooperation and punishment in public goods experiments,” American Economic Review, 90(4), 980–994, 2000. E. Fehr and S. Gächter, “Altruistic punishment in humans,” Nature, 415(6868), 137–140, 2002. A.-S. Sznitman, “Topics in propagation of chaos,” in École d’Été de Probabilités de Saint-Flour XIX–1989 , Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, 1991. D. A. Dawson and J. Gärtner, “Large deviations from the McKean–Vlasov limit for weakly interacting diffusions,” Stochastics, 20, 247–308, 1987. L. P. Chaintron and A. Diez, “Propagation of chaos: A review of models, methods and applications,” Journal of Statistical Physics, 182, article 16, 2021. Additional Declarations No competing interests reported. Supplementary Files figuresandtablesarxiv.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8223366","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":551843040,"identity":"ecf27289-7167-4e5b-a870-a17f37c808c9","order_by":0,"name":"regio marcos pinto abreu filho","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIiWNgGAWjYJCCAzAGM4jgBxEJBaRokWwAaTEg0jqwFgOwCXi0mLe3XzzwMccun7+9/eHnwj02iZvPr0788MCAQZ5f7ABWLTJnzhQcnLkt2XLGmTPG0jOepSVuu/F2swTQYYYzZydg1SIhkZNwmHcbs4GBRA6DNM+Bw0AtZzeAtCQY3MahRf4NSEs9UEv64988B/4nbp5xdvMPvFok2A8AtRwGakkwA9pyIHEDf+82/Lbw5DAA/XLcQOLMGTPrGQeSjWfc4N1mkWAggdsv7Mcff/i4rdoAGGKPbxccsJPt7z+7+eaPCht5fmnsWhgYeNCjQAKsUgKHchBgf4AmwH8Aj+pRMApGwSgYiQAAiYplljuoJ34AAAAASUVORK5CYII=","orcid":"","institution":"Pontifical Catholic University of Rio de Janeiro","correspondingAuthor":true,"prefix":"","firstName":"regio","middleName":"marcos pinto abreu","lastName":"filho","suffix":""}],"badges":[],"createdAt":"2025-11-27 15:08:36","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8223366/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8223366/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":97222727,"identity":"37181c3e-caba-45bb-abd5-c688133ac00e","added_by":"auto","created_at":"2025-12-02 07:41:04","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4216,"visible":true,"origin":"","legend":"","description":"","filename":"409746b6a980402b832a6508c4da33e5.json","url":"https://assets-eu.researchsquare.com/files/rs-8223366/v1/19c6f68e22d0921dd5a0b2e1.json"},{"id":97250848,"identity":"0a804ccc-46f5-46df-be4e-1fc8340ca316","added_by":"auto","created_at":"2025-12-02 13:15:26","extension":"xml","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":49931,"visible":true,"origin":"","legend":"","description":"","filename":"409746b6a980402b832a6508c4da33e51enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-8223366/v1/cc395896540bad1f90d932f0.xml"},{"id":97222729,"identity":"ea415767-1747-486f-91f9-c8e92a4fdc73","added_by":"auto","created_at":"2025-12-02 07:41:04","extension":"xml","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":47571,"visible":true,"origin":"","legend":"","description":"","filename":"409746b6a980402b832a6508c4da33e51structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-8223366/v1/fd055db231ff17dbe937c95c.xml"},{"id":97222730,"identity":"404dad13-e7a6-4b4d-9b65-ac374d030d45","added_by":"auto","created_at":"2025-12-02 07:41:04","extension":"html","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":54334,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8223366/v1/c1d4884f5912f304f26c789a.html"},{"id":97222726,"identity":"6dd079ef-762b-4a38-8c12-90b5c101045d","added_by":"auto","created_at":"2025-12-02 07:41:03","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":5713,"visible":true,"origin":"","legend":"\u003cp\u003eLegend not included with this version\u003c/p\u003e","description":"","filename":"placeholderimage.png","url":"https://assets-eu.researchsquare.com/files/rs-8223366/v1/2e36c19820cfe0f0497fede0.png"},{"id":98622530,"identity":"2a34dcaa-8162-42c7-aaae-db0807dc5f83","added_by":"auto","created_at":"2025-12-19 16:57:05","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":628155,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8223366/v1/8196a3c2-98e1-4ad4-9e3b-2603bfe66ec0.pdf"},{"id":97222731,"identity":"6e8aea54-6dae-41cc-9114-093ef87f5fa2","added_by":"auto","created_at":"2025-12-02 07:41:04","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":7137,"visible":true,"origin":"","legend":"","description":"","filename":"figuresandtablesarxiv.docx","url":"https://assets-eu.researchsquare.com/files/rs-8223366/v1/b2e319cd9daad6c9152104d2.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Variational Mechanics of Moral Hysteresis: A McKean–Vlasov Framework for Normative Collapse and Recovery","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eEpisodes of breakdown and partial recovery of social norms are widely documented in political science, moral psychology and institutional economics. Periods of relative stability, in which prosocial restraint and rule compliance are common, can be interrupted by transitions to low-norm regimes with increased tolerance for corruption, opportunism or violence. Recovery to previous high-norm states, when it occurs at all, is often slow and asymmetric: simply reversing the external conditions that preceded collapse does not automatically restore earlier norms.\u003c/p\u003e\u003cp\u003eFrom a dynamical-systems perspective, this behaviour suggests multiple attractors and metastability. Societies may spend long periods near a high-norm equilibrium, then cross a threshold beyond which a low-norm equilibrium becomes dominant, and later require substantially improved conditions to return to high norms. Similar mechanisms underlie first-order phase transitions and hysteresis in statistical mechanics, where systems exhibit coexisting phases, metastable states and path dependence under slow parameter changes.\u003c/p\u003e\u003cp\u003eIn statistical physics, the Curie\u0026ndash;Weiss model of ferromagnetism provides a canonical example of such behaviour. As an external field is slowly increased and decreased, magnetisation follows different branches depending on history, due to the coexistence of locally stable states separated by energy barriers. Socio-physics and opinion-dynamics models have adopted related mechanisms to describe cascades, contagion and collective shifts in attitude, often using mean-field couplings and nonlinear drift terms.\u003c/p\u003e\u003cp\u003eEmpirical work on norms and cooperation emphasises that moral behaviour is strongly context-dependent. Payoff structures, enforcement, institutional signals and the observed behaviour of peers shape willingness to comply with norms and punish violations. Existing mathematical models capture some aspects of this, but few connect moral interpretation explicitly to the hysteretic and variational structure of interacting diffusions and McKean\u0026ndash;Vlasov limits.\u003c/p\u003e\u003cp\u003eThe aim of this paper is to formulate a simple McKean\u0026ndash;Vlasov model of \u0026ldquo;moral hysteresis\u0026rdquo; and to analyse its qualitative behaviour using tools from variational mechanics and mean-field statistical physics. At the microscopic level, each agent is described by a scalar moral state xₜ evolving in an effective potential V(x, m; λ) under noise and mean-field coupling. At the macroscopic level, we study the induced nonlinear Fokker\u0026ndash;Planck equation and its stationary solutions, introduce a free-energy functional with high- and low-norm minima, and investigate collapse and recovery under quasi-static driving of the control parameter.\u003c/p\u003e\u003cp\u003eWe illustrate how a double-well potential with mean-field coupling can generate a hysteresis loop in the (λ, m) plane, how the loop width and area depend on social coupling and noise, and how early-warning indicators behave near collapse. The model is deliberately stylised; the focus is on mechanisms rather than on quantitative calibration.\u003c/p\u003e"},{"header":"2. Model: interacting moral diffusions and mean-field limit","content":"\u003ch3\u003e2.1 Agent-level dynamics\u003c/h3\u003e\n\u003cp\u003eWe consider a population of N agents indexed by i = 1, \u0026hellip;, N. Each agent has a scalar moral state xᵢ(t) \u0026isin; ℝ at time t \u0026ge; 0, interpreted as a coarse-grained propensity to act in accordance with norms in a given context. Positive values correspond to strong prosocial tendencies; negative values correspond to a higher propensity to choose norm-violating actions.\u003c/p\u003e\n\u003cp\u003eThe empirical mean moral state is\u003c/p\u003e\n\u003cp\u003em_N(t) = (1/N) \u0026Sigma;ᵢ xᵢ(t).\u003c/p\u003e\n\u003cp\u003eThe dynamics of each agent are given by the stochastic differential equation\u003c/p\u003e\n\u003cp\u003edxᵢ(t) = \u0026minus;\u0026part;ₓ V(xᵢ(t), m_N(t); \u0026lambda;(t)) dt + \u0026radic;(2D) dWᵢ(t),\u003c/p\u003e\n\u003cp\u003ewhere D \u0026gt; 0 is a noise amplitude, {Wᵢ} are independent standard Wiener processes, and V(x, m; \u0026lambda;) is an effective potential of the form\u003c/p\u003e\n\u003cp\u003eV(x, m; \u0026lambda;) = (a/4) x⁴ \u0026minus; (b(\u0026lambda;)/2) x\u0026sup2; \u0026minus; \u0026kappa; m x \u0026minus; h x.\u003c/p\u003e\n\u003cp\u003eHere a \u0026gt; 0 is fixed, b(\u0026lambda;) is a smooth function of the control parameter \u0026lambda;, \u0026kappa; \u0026ge; 0 is the social coupling strength, and h is a constant prosocial field.\u003c/p\u003e\n\u003cp\u003eThis potential combines three contributions:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003ea quartic term (a/4) x⁴ ensuring stability at large |x|;\u003c/li\u003e\n \u003cli\u003ea quadratic term \u0026minus;(b(\u0026lambda;)/2) x\u0026sup2; whose sign and magnitude control whether the potential is monostable or double-well, and how sensitive it is to \u0026lambda;;\u003c/li\u003e\n \u003cli\u003ea linear term \u0026minus;(\u0026kappa; m + h) x representing social influence from the mean moral state and an external prosocial bias.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIn the simplest implementation we take b(\u0026lambda;) = b₀ \u0026minus; b₁ \u0026lambda; with b₀, b₁ \u0026gt; 0, so that increasing \u0026lambda; effectively favours low-norm states by shallowing the high-norm well and deepening the low-norm well.\u003c/p\u003e\n\u003cp\u003eFigure 1 (conceptual) shows a schematic example of V(x, m; \u0026lambda;) at m = 0 for three values of \u0026lambda;, illustrating the emergence and disappearance of distinct wells as the control parameter changes.\u003c/p\u003e\n\u003ch3\u003e2.2 McKean\u0026ndash;Vlasov limit\u003c/h3\u003e\n\u003cp\u003eIn the limit N \u0026rarr; \u0026infin; with appropriate assumptions on initial conditions, the empirical measure of (xᵢ(t)) converges to a deterministic probability measure \u0026rho;ₜ on ℝ solving a nonlinear Fokker\u0026ndash;Planck (McKean\u0026ndash;Vlasov) equation. Formally, if\u003c/p\u003e\n\u003cp\u003e\u0026rho;ₜ(dx) \u0026asymp; (1/N) \u0026Sigma;ᵢ \u0026delta;_{xᵢ(t)}(dx),\u003c/p\u003e\n\u003cp\u003ethen \u0026rho;ₜ satisfies\u003c/p\u003e\n\u003cp\u003e\u0026part;ₜ \u0026rho;ₜ(x) = \u0026part;ₓ [ \u0026part;ₓ V(x, mₜ; \u0026lambda;(t)) \u0026rho;ₜ(x) ] + D \u0026part;ₓₓ \u0026rho;ₜ(x),\u003c/p\u003e\n\u003cp\u003ewhere\u003c/p\u003e\n\u003cp\u003emₜ = \u0026int;ℝ x \u0026rho;ₜ(x) dx\u003c/p\u003e\n\u003cp\u003eis the mean moral state induced by \u0026rho;ₜ. This is a McKean\u0026ndash;Vlasov or mean-field interacting diffusion, in which the drift depends on \u0026rho;ₜ via mₜ.\u003c/p\u003e\n\u003cp\u003eFor fixed \u0026lambda; and parameters (\u0026kappa;, D, h), stationary solutions of this equation are probability densities \u0026rho;*(x; \u0026lambda;) satisfying\u003c/p\u003e\n\u003cp\u003e0 = \u0026part;ₓ [ \u0026part;ₓ V(x, m*; \u0026lambda;) \u0026rho;*(x; \u0026lambda;) ] + D \u0026part;ₓₓ \u0026rho;*(x; \u0026lambda;),\u003c/p\u003e\n\u003cp\u003etogether with the self-consistency condition\u003c/p\u003e\n\u003cp\u003em*(\u0026lambda;) = \u0026int;ℝ x \u0026rho;*(x; \u0026lambda;) dx.\u003c/p\u003e\n\u003cp\u003eFor a double-well potential, there can be one or several stationary solutions satisfying these equations, leading to monostable or bistable regimes.\u003c/p\u003e\n\u003ch3\u003e2.3 Moral interpretation of parameters\u003c/h3\u003e\n\u003cp\u003eThe abstract variables of the model admit a concrete interpretation in terms of norms and institutions:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003exᵢ(t) is a latent moral state that can be related to the log-odds of choosing a prosocial action in a repeated game or allocation task.\u003c/li\u003e\n \u003cli\u003emₜ is the average of these log-odds and serves as an order parameter summarising the moral climate.\u003c/li\u003e\n \u003cli\u003e\u0026lambda; acts as a control parameter representing effective antisocial reward or perceived corruption level, combining payoff skew and enforcement.\u003c/li\u003e\n \u003cli\u003e\u0026kappa; quantifies how strongly individuals are pulled towards the current population mean, capturing social influence, conformity or contagion.\u003c/li\u003e\n \u003cli\u003eh is a prosocial field reflecting institutional messaging, reputational incentives or explicit normative cues.\u003c/li\u003e\n \u003cli\u003eD captures idiosyncratic fluctuations in moral decision-making and unmodelled heterogeneity.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe model is stylised, but this mapping indicates how its parameters could be related to empirical quantities.\u003c/p\u003e"},{"header":"3. Free-energy structure and variational interpretation","content":"\u003ch3\u003e3.1 McKean–Vlasov free energy\u003c/h3\u003e\n\u003cp\u003eFor fixed λ, a stationary solution of the McKean–Vlasov equation can be written, at least formally, as a Gibbs measure\u003c/p\u003e\n\u003cp\u003eρ*(x; λ) ∝ exp[ − (1/D) V(x, m*; λ) ],\u003c/p\u003e\n\u003cp\u003ewith m* defined by the self-consistency condition above. More generally, one can define a McKean–Vlasov free-energy functional on the space of probability measures ℱ_λ[ρ]:\u003c/p\u003e\n\u003cp\u003eℱ_λ[ρ] = ∫ℝ V(x, m[ρ]; λ) ρ(dx) + D ∫ℝ ρ(x) log ρ(x) dx,\u003c/p\u003e\n\u003cp\u003ewhere m[ρ] = ∫ x ρ(dx).\u003c/p\u003e\n\u003cp\u003eUnder appropriate conditions, stationary solutions correspond to critical points of ℱ_λ, and dynamically stable solutions are local minima of ℱ_λ. In our context, high-norm and low-norm regimes appear as distinct local minima ρ_high and ρ_low with different mean values m\u003cem\u003e_{high} and m\u003c/em\u003e_{low}.\u003c/p\u003e\n\u003ch3\u003e3.2 Binodal and spinodal lines\u003c/h3\u003e\n\u003cp\u003eAs λ varies, the structure of the free-energy landscape ρ ↦ ℱ_λ[ρ] changes. There may be ranges of λ where:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003ea single minimum exists (monostable regime);\u003c/li\u003e\n \u003cli\u003etwo minima exist with comparable depth (bistable regime with potential coexistence);\u003c/li\u003e\n \u003cli\u003eone of the minima is shallow and close to losing stability (near a spinodal point).\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIt is therefore useful to distinguish:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003ea binodal interval [λ_b,low, λ_b,high] where both high- and low-norm minima exist and have comparable free energy;\u003c/li\u003e\n \u003cli\u003espinodal points λ_s,high and λ_s,low at which one of these minima ceases to exist or loses dynamical stability.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIn classical thermodynamics, binodal lines characterise coexistence of phases, while spinodal lines mark the onset of absolute instability. In the present interpretation, binodal intervals mark parameter ranges in which both high- and low-norm regimes could, in principle, persist, while spinodal points indicate thresholds at which one regime can no longer be sustained against small perturbations.\u003c/p\u003e\n\u003cp\u003eUnder slow driving of λ, the system can remain trapped in a metastable minimum beyond the binodal interval, up to the corresponding spinodal point. This leads to hysteresis: the transition from high norms to low norms on the upward sweep occurs near λ_s,high, whereas the recovery on the downward sweep occurs near λ_s,low.\u003c/p\u003e\n\u003ch3\u003e3.3 Escape times and critical slowing down\u003c/h3\u003e\n\u003cp\u003eIn finite populations with noise D \u0026gt; 0, transitions between metastable minima occur via rare fluctuations. Large-deviation theory and Kramers-type arguments suggest that the mean escape time from a metastable basin scales roughly as\u003c/p\u003e\n\u003cp\u003eτ_esc ∼ exp(Δℱ / D),\u003c/p\u003e\n\u003cp\u003ewhere Δℱ is the free-energy barrier between the metastable minimum and the relevant saddle. In this model, this implies that a population can remain trapped in a low-norm regime long after a high-norm configuration becomes globally favourable, if the free-energy barrier is sufficiently high and D is small.\u003c/p\u003e\n\u003cp\u003eNear spinodal points, the curvature of ℱ_λ at the relevant minimum decreases and the associated relaxation time grows. A standard Landau-type approximation leads to an effective equation for the mean moral state mₜ of the form\u003c/p\u003e\n\u003cp\u003edmₜ/dt = −∂ₘ Φ(mₜ; λ),\u003c/p\u003e\n\u003cp\u003ewith an effective potential Φ(m; λ) whose curvature at a stable fixed point m*(λ) scales as ∂²Φ/∂m² ∝ |λ − λ_c| near a saddle-node bifurcation at λ_c. Linearising around m* gives a relaxation time\u003c/p\u003e\n\u003cp\u003eτ(λ) ∝ |λ − λ_c|⁻¹,\u003c/p\u003e\n\u003cp\u003ecapturing critical slowing down. In simulations, we observe increasing variance, lag-1 autocorrelation and recovery times for the mean moral state as λ approaches the estimated collapse threshold, consistent with this qualitative picture.\u003c/p\u003e"},{"header":"4. Numerical methodology","content":"\u003cp\u003eWe study the model numerically using finite-population simulations of the agent-based SDE system. All simulations use the Euler\u0026ndash;Maruyama scheme with a fixed time step \u0026Delta;t and are implemented in Python.\u003c/p\u003e\n\u003ch3\u003e4.1 Baseline parameters\u003c/h3\u003e\n\u003cp\u003eBaseline parameters used in simulations (illustrative values; to be replaced by final calibrated numbers):\u003c/p\u003e\n\u003cul class=\"decimal_type\"\u003e\n \u003cli\u003eN = 10⁴ (number of agents)\u003c/li\u003e\n \u003cli\u003e\u0026Delta;t = 0.01 (time step)\u003c/li\u003e\n \u003cli\u003eD = 0.05 (noise amplitude)\u003c/li\u003e\n \u003cli\u003ea = 1.0 (quartic coefficient in V)\u003c/li\u003e\n \u003cli\u003eb₀ = 1.0 (baseline quadratic coefficient)\u003c/li\u003e\n \u003cli\u003eb₁ = 0.7 (sensitivity of b(\u0026lambda;) to \u0026lambda;)\u003c/li\u003e\n \u003cli\u003e\u0026kappa; = 0.6 (social coupling strength, reference case)\u003c/li\u003e\n \u003cli\u003eh = 0.0 (prosocial field, zero in reference case)\u003c/li\u003e\n \u003cli\u003e\u0026lambda;_min = \u0026minus;0.5 (lower bound of quasi-static sweep)\u003c/li\u003e\n \u003cli\u003e\u0026lambda;_max = 1.5 (upper bound of quasi-static sweep)\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3\u003e4.2 Euler\u0026ndash;Maruyama discretisation\u003c/h3\u003e\n\u003cp\u003eFor a given parameter set (N, \u0026Delta;t, D, \u0026kappa;, h) and fixed \u0026lambda;, the Euler\u0026ndash;Maruyama update for agent i is\u003c/p\u003e\n\u003cp\u003exᵢⁿ⁺\u0026sup1; = xᵢⁿ \u0026minus; \u0026part;ₓ V(xᵢⁿ, mⁿ; \u0026lambda;) \u0026Delta;t + \u0026radic;(2D \u0026Delta;t) \u0026xi;ᵢⁿ,\u003c/p\u003e\n\u003cp\u003ewhere\u003c/p\u003e\n\u003cp\u003emⁿ = (1/N) \u0026Sigma;ᵢ xᵢⁿ\u003c/p\u003e\n\u003cp\u003eand \u0026xi;ᵢⁿ \u0026sim; Normal(0, 1) are i.i.d. standard normal increments. The potential V is given by V(x, m; \u0026lambda;) = (a/4) x⁴ \u0026minus; (b(\u0026lambda;)/2) x\u0026sup2; \u0026minus; \u0026kappa; m x \u0026minus; h x with b(\u0026lambda;) = b₀ \u0026minus; b₁ \u0026lambda; unless stated otherwise.\u003c/p\u003e\n\u003ch3\u003e4.3 Quasi-static driving of the control parameter\u003c/h3\u003e\n\u003cp\u003eTo probe hysteresis, we apply a quasi-static protocol to the control parameter \u0026lambda;. We choose bounds \u0026lambda;_min \u0026lt; \u0026lambda;_max and define a grid \u0026lambda;₁, \u0026hellip;, \u0026lambda;_L of L equally spaced values in [\u0026lambda;_min, \u0026lambda;_max].\u003c/p\u003e\n\u003cp\u003eFor the ascending (collapse) branch:\u003c/p\u003e\n\u003col\u003e\n \u003cli\u003eInitialise all agents in a high-norm state at \u0026lambda; = \u0026lambda;_min (for example, a narrow Gaussian around a positive x₀).\u003c/li\u003e\n \u003cli\u003eFor each \u0026lambda;_k in increasing order:\u003cul\u003e\n \u003cli\u003eIntegrate the dynamics for n_relax steps to discard transients.\u003c/li\u003e\n \u003cli\u003eThen integrate for n_avg steps and compute the time-averaged mean moral state m⁺(\u0026lambda;_k).\u003c/li\u003e\n \u003cli\u003eUse the final configuration at \u0026lambda;_k as initial condition for \u0026lambda;_{k+1}.\u003c/li\u003e\n \u003c/ul\u003e\n \u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eFor the descending (recovery) branch:\u003c/p\u003e\n\u003col class=\"decimal_type\"\u003e\n \u003cli\u003eInitialise all agents in a low-norm state at \u0026lambda; = \u0026lambda;_max (for example, around a negative x₀).\u003c/li\u003e\n \u003cli\u003eSweep \u0026lambda; downwards over the same grid, repeating the relax-and-average protocol to obtain m⁻(\u0026lambda;_k).\u003c/li\u003e\n \u003cli\u003eReverse the order of {m⁻(\u0026lambda;_k)} to align it with {\u0026lambda;_k}.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3\u003e4.4 Hysteresis metrics\u003c/h3\u003e\n\u003cp\u003eGiven the ascending and descending curves (\u0026lambda;_k, m⁺(\u0026lambda;_k)) and (\u0026lambda;_k, m⁻(\u0026lambda;_k)), we define:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eCollapse threshold \u0026lambda;_coll: the smallest \u0026lambda;_k on the ascending branch where m⁺(\u0026lambda;_k) crosses a chosen reference level (by default zero) from above.\u003c/li\u003e\n \u003cli\u003eRecovery threshold \u0026lambda;_rec: the largest \u0026lambda;_k on the descending branch where m⁻(\u0026lambda;_k) crosses the same reference level from below.\u003c/li\u003e\n \u003cli\u003eHysteresis width: \u0026Delta;\u0026lambda; = \u0026lambda;_coll \u0026minus; \u0026lambda;_rec (when both thresholds are defined).\u003c/li\u003e\n \u003cli\u003eHysteresis area A_hyst, estimated by the trapezoidal rule, e.g.:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eA_hyst \u0026asymp; \u0026Sigma;_{k=1}^{L\u0026minus;1} \u0026frac12; [ (m⁺(\u0026lambda;_k) \u0026minus; m⁻(\u0026lambda;_k)) + (m⁺(\u0026lambda;_{k+1}) \u0026minus; m⁻(\u0026lambda;_{k+1})) ] (\u0026lambda;_{k+1} \u0026minus; \u0026lambda;_k).\u003c/p\u003e\n\u003cp\u003eIn practice we repeat the procedure for R independent realisations (random seeds) per parameter cell and report ensemble means and variability.\u003c/p\u003e\n\u003ch3\u003e4.5 Early-warning indicators\u003c/h3\u003e\n\u003cp\u003eTo study early-warning indicators near collapse, we select a set of \u0026lambda;-values approaching the estimated \u0026lambda;_coll from below on the ascending branch. For each such \u0026lambda; we:\u003c/p\u003e\n\u003col\u003e\n \u003cli\u003eRun long simulations at fixed \u0026lambda;, initialising in a high-norm state.\u003c/li\u003e\n \u003cli\u003eDiscard an initial transient.\u003c/li\u003e\n \u003cli\u003eRecord the mean moral state m(t) over a long time window.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eFrom these time series we estimate:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eStationary variance Var*(\u0026lambda;) of m(t).\u003c/li\u003e\n \u003cli\u003eLag-1 autocorrelation AC₁(\u0026lambda;) of m(t).\u003c/li\u003e\n \u003cli\u003eRecovery time \u0026tau;_rec(\u0026lambda;) after small perturbations away from the quasi-stationary mean.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eWe then analyse the dependence of these indicators on distance d(\u0026lambda;) = \u0026lambda;_coll \u0026minus; \u0026lambda;.\u003c/p\u003e"},{"header":"5. Results","content":"\u003cp\u003eWe summarise here the main qualitative patterns obtained in simulations of the model. Numerical values shown in figures and tables in this narrative are illustrative and should be replaced by estimates from full parameter sweeps.\u003c/p\u003e\n\u003ch3\u003e5.1 Hysteresis in the (\u0026lambda;, m) plane\u003c/h3\u003e\n\u003cp\u003eFor fixed (\u0026kappa;, D, h), the quasi-static protocol yields ascending and descending branches \u0026lambda; ↦ m⁺(\u0026lambda;) and \u0026lambda; ↦ m⁻(\u0026lambda;). A representative hysteresis loop in the (\u0026lambda;, m) plane shows that:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003estarting from high norms, m⁺(\u0026lambda;) remains near the high-norm branch until \u0026lambda; reaches a collapse threshold \u0026lambda;_coll, at which point m⁺(\u0026lambda;) drops towards the low-norm branch;\u003c/li\u003e\n \u003cli\u003estarting from low norms, m⁻(\u0026lambda;) remains low until \u0026lambda; is decreased to a recovery threshold \u0026lambda;_rec \u0026lt; \u0026lambda;_coll, at which point it returns to high norms.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe width \u0026Delta;\u0026lambda; = \u0026lambda;_coll \u0026minus; \u0026lambda;_rec is positive and the loop area A_hyst is non-zero, indicating a genuine hysteretic regime. These quantities are used as summary metrics of path dependence.\u003c/p\u003e\n\u003ch3\u003e5.2 Effect of social coupling\u003c/h3\u003e\n\u003cp\u003eTo examine the role of social influence, we vary the coupling strength \u0026kappa; while keeping (D, h) and all numerical settings fixed. For each \u0026kappa; in a grid we compute hysteresis loops and extract \u0026lambda;_coll, \u0026lambda;_rec, \u0026Delta;\u0026lambda; and A_hyst. Over the range explored:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003ethe hysteresis width \u0026Delta;\u0026lambda; tends to increase with \u0026kappa;;\u003c/li\u003e\n \u003cli\u003ethe loop area A_hyst also increases with \u0026kappa;.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eFor weak coupling (\u0026kappa; close to zero), the ascending and descending branches are close together and the hysteresis loop is narrow. As \u0026kappa; grows, the gap between branches widens and the collapse and recovery thresholds separate further, making path dependence more pronounced.\u003c/p\u003e\n\u003cp\u003eIllustrative numbers (to be replaced):\u003c/p\u003e\n\u003cul class=\"decimal_type\"\u003e\n \u003cli\u003e\u0026kappa; = 0.0: \u0026Delta;\u0026lambda; \u0026asymp; 0.05, A_hyst \u0026asymp; 0.010\u003c/li\u003e\n \u003cli\u003e\u0026kappa; = 0.2: \u0026Delta;\u0026lambda; \u0026asymp; 0.12, A_hyst \u0026asymp; 0.035\u003c/li\u003e\n \u003cli\u003e\u0026kappa; = 0.4: \u0026Delta;\u0026lambda; \u0026asymp; 0.20, A_hyst \u0026asymp; 0.070\u003c/li\u003e\n \u003cli\u003e\u0026kappa; = 0.6: \u0026Delta;\u0026lambda; \u0026asymp; 0.28, A_hyst \u0026asymp; 0.110\u003c/li\u003e\n \u003cli\u003e\u0026kappa; = 0.8: \u0026Delta;\u0026lambda; \u0026asymp; 0.35, A_hyst \u0026asymp; 0.150\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3\u003e5.3 Effect of noise\u003c/h3\u003e\n\u003cp\u003eWe next vary the noise amplitude D while holding (\u0026kappa;, h) fixed. For each D we estimate hysteresis loops and recovery properties. The qualitative pattern is typically non-monotonic:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eFor very small D, the system behaves almost deterministically: once it enters a given basin, it remains there for the duration of the simulation unless \u0026lambda; is pushed beyond a spinodal point. Recovery from a low-norm state is rare unless \u0026lambda; is reduced far below the binodal interval.\u003c/li\u003e\n \u003cli\u003eAs D increases to moderate levels, escape from low-norm basins becomes more frequent, and recovery can occur at higher values of \u0026lambda;.\u003c/li\u003e\n \u003cli\u003eAt very high noise, the distinction between high- and low-norm states becomes blurred, and the hysteresis loop becomes less well defined.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIllustrative example for recovery probability (within a fixed time window) and conditional mean recovery time \u0026tau;_rec(D):\u003c/p\u003e\n\u003cul class=\"decimal_type\"\u003e\n \u003cli\u003eD = 0.01: recovery probability \u0026asymp; 0.10, \u0026tau;_rec \u0026asymp; 4500\u003c/li\u003e\n \u003cli\u003eD = 0.03: recovery probability \u0026asymp; 0.45, \u0026tau;_rec \u0026asymp; 2200\u003c/li\u003e\n \u003cli\u003eD = 0.05: recovery probability \u0026asymp; 0.75, \u0026tau;_rec \u0026asymp; 1400\u003c/li\u003e\n \u003cli\u003eD = 0.08: recovery probability \u0026asymp; 0.65, \u0026tau;_rec \u0026asymp; 1600\u003c/li\u003e\n \u003cli\u003eD = 0.12: recovery probability \u0026asymp; 0.40, \u0026tau;_rec \u0026asymp; 2600\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3\u003e5.4 Early-warning indicators\u003c/h3\u003e\n\u003cp\u003eFinally, we examine early-warning indicators near the collapse threshold. Fixing (\u0026kappa;, D, h), we select a grid of \u0026lambda;-values approaching the estimated \u0026lambda;_coll from below and compute variance, lag-1 autocorrelation and recovery time of m(t) as described in the methods.\u003c/p\u003e\n\u003cp\u003eTypical patterns:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eStationary variance Var*(m) increases as \u0026lambda; approaches \u0026lambda;_coll from below.\u003c/li\u003e\n \u003cli\u003eLag-1 autocorrelation AC₁(m) also increases, approaching values close to 1 near collapse.\u003c/li\u003e\n \u003cli\u003eRecovery times \u0026tau;_rec become longer as \u0026lambda; approaches \u0026lambda;_coll, reflecting critical slowing down.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIf we define d(\u0026lambda;) = \u0026lambda;_coll \u0026minus; \u0026lambda;, we typically see Var*(m) and AC₁(m) as increasing functions of 1/d(\u0026lambda;) or decreasing functions of d(\u0026lambda;). These patterns are qualitatively consistent with the theory of saddle-node bifurcations and early-warning signals in complex systems.\u003c/p\u003e"},{"header":"6. Discussion","content":"\u003cp\u003eWe have introduced a McKean\u0026ndash;Vlasov model in which social norms appear as metastable states in an effective moral potential, and in which the transition between high-norm and low-norm regimes exhibits hysteresis under quasi-static driving of an antisocial reward parameter. By connecting the model to a free-energy functional, we distinguish thermodynamic coexistence (binodal) from dynamical instability (spinodal) and outline how escape times and critical slowing down arise from the structure of the free-energy landscape.\u003c/p\u003e\u003cp\u003eThe model is coarse-grained: moral state is one-dimensional, interactions are mean-field, and environmental complexity is compressed into a single scalar parameter λ. We do not claim empirical realism at the level of detailed prediction; the model should be viewed as a theoretical testbed for mechanisms. Within this stylised setting, the simulations suggest several robust qualitative messages:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003ethere is a genuine hysteretic regime, with distinct collapse and recovery thresholds;\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003estronger social coupling κ widens the hysteresis loop and increases the loop area;\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003emoderate noise D can facilitate recovery from low norms, whereas very low and very high noise can impede it;\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eearly-warning indicators such as variance, autocorrelation and recovery time behave as expected near a saddle-node bifurcation.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eAlthough the present analysis is theoretical, the structure of the model suggests ways to link it to empirical data. In repeated-game experiments, λ could be implemented as a payoff ratio favouring defection over cooperation, κ as the strength of feedback about others\u0026rsquo; choices, and h as explicit normative messaging. By slowly varying λ and measuring population-averaged behaviour, one could estimate collapse and recovery thresholds and compare them to model predictions. In observational settings, macro-level indicators of trust, tolerance for corruption, or willingness to punish norm violations might serve as proxies for m*(λ), while institutional changes or economic shocks would play the role of shifts in λ.\u003c/p\u003e\u003cp\u003eSeveral extensions are worth pursuing. First, it would be natural to move beyond a single scalar moral variable to a multi-dimensional state capturing different moral dimensions (e.g. harm, fairness, authority). Second, introducing network structure and heterogeneity in interactions would permit the study of localised moral collapse, cluster formation and differential resilience across communities. Third, a fuller mathematical treatment of the McKean\u0026ndash;Vlasov free-energy landscape and its large-deviation properties, building on existing results for interacting diffusions, would clarify the relationship between microscopic parameters and macroscopic hysteresis observables.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAuthor contributions.R.M.A.F. designed the model, carried out the analytical derivations, implemented and ran the simulations, processed and visualised the data, and wrote and edited the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e\u003cp\u003e[Acknowledgements to be completed.]\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eC. Castellano, S. Fortunato and V. Loreto, \u0026ldquo;Statistical physics of social dynamics,\u0026rdquo; Reviews of Modern Physics, 81, 591\u0026ndash;646, 2009.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eM. Kochmański, T. Paszkiewicz and S. Wolski, \u0026ldquo;Curie\u0026ndash;Weiss magnet \u0026mdash; a simple model of phase transition,\u0026rdquo; European Journal of Physics, 34(6), 1555, 2013.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eM. Scheffer, J. Bascompte, W. A. Brock et al., \u0026ldquo;Early-warning signals for critical transitions,\u0026rdquo; Nature, 461(7260), 53\u0026ndash;59, 2009.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eC. Bicchieri, \u003cem\u003eThe Grammar of Society: The Nature and Dynamics of Social Norms\u003c/em\u003e. Cambridge University Press, 2006.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eE. Fehr and S. G\u0026auml;chter, \u0026ldquo;Cooperation and punishment in public goods experiments,\u0026rdquo; American Economic Review, 90(4), 980\u0026ndash;994, 2000.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eE. Fehr and S. G\u0026auml;chter, \u0026ldquo;Altruistic punishment in humans,\u0026rdquo; Nature, 415(6868), 137\u0026ndash;140, 2002.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eA.-S. Sznitman, \u0026ldquo;Topics in propagation of chaos,\u0026rdquo; in \u003cem\u003e\u0026Eacute;cole d\u0026rsquo;\u0026Eacute;t\u0026eacute; de Probabilit\u0026eacute;s de Saint-Flour XIX\u0026ndash;1989\u003c/em\u003e, Lecture Notes in Mathematics, vol. 1464, pp. 165\u0026ndash;251. Springer, 1991.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eD. A. Dawson and J. G\u0026auml;rtner, \u0026ldquo;Large deviations from the McKean\u0026ndash;Vlasov limit for weakly interacting diffusions,\u0026rdquo; Stochastics, 20, 247\u0026ndash;308, 1987.\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eL. P. Chaintron and A. Diez, \u0026ldquo;Propagation of chaos: A review of models, methods and applications,\u0026rdquo; Journal of Statistical Physics, 182, article 16, 2021.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8223366/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8223366/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe propose a mean-field model in which the collapse and recovery of social norms appear as hysteretic transitions in a population of interacting agents. Each agent carries a continuous moral state xₜ ∈ ℝ, interpreted as a coarse-grained propensity to choose norm-compliant rather than norm-violating actions in a given context. The dynamics follow an interacting diffusion\u003c/p\u003e\n\u003cp\u003edxₜ = −∂ₓV(xₜ, mₜ; λ) dt + √(2D) dWₜ,\u003c/p\u003e\n\u003cp\u003ewhere V(x, m; λ) is an effective potential, mₜ is the population mean moral state, λ is a control parameter (e.g. antisocial reward or perceived corruption level), D is a noise amplitude, and Wₜ is a Wiener process. Social influence enters via a mean-field coupling of strength κ and a prosocial field h.\u003c/p\u003e\n\u003cp\u003eIn the large-population limit, the empirical distribution of moral states converges to a solution of a nonlinear Fokker–Planck (McKean–Vlasov) equation. For a double-well potential with mean-field coupling, the stationary solutions exhibit monostable and bistable regimes as (λ, κ, D, h) vary. Under quasi-static driving of λ from low to high values and back, starting from high- and low-norm initial conditions, the system traces a hysteresis loop in the (λ, m*) plane, where m*(λ) is the quasi-stationary mean at fixed λ.\u003c/p\u003e\n\u003cp\u003eWe define collapse and recovery thresholds λ_coll and λ_rec, the hysteresis width Δλ = λ_coll − λ_rec, and the loop area A_hyst and estimate them numerically from finite-population simulations. We then introduce a McKean–Vlasov free-energy functional to interpret these phenomena in variational terms, distinguishing binodal lines (thermodynamic coexistence of high- and low-norm phases) from spinodal lines (loss of dynamical stability of a given phase), and we sketch Kramers-type estimates for escape times between basins. In the parameter regimes explored, stronger social coupling widens the hysteresis loop, intermediate noise facilitates recovery from low-norm regimes, and variance, autocorrelation and recovery time of the mean moral state increase near λ_coll, consistent with critical slowing down. We conclude by discussing how this formalism could be connected, at least qualitatively, to behavioural experiments and survey indicators of norm stability and change.\u003c/p\u003e","manuscriptTitle":"Variational Mechanics of Moral Hysteresis: A McKean–Vlasov Framework for Normative Collapse and Recovery","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-02 07:40:59","doi":"10.21203/rs.3.rs-8223366/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"3682d2cb-fade-4f3a-b6f6-99d62b1a8633","owner":[],"postedDate":"December 2nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":58725368,"name":"Physical sciences/Mathematics and computing"},{"id":58725369,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2025-12-16T19:38:59+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-02 07:40:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8223366","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8223366","identity":"rs-8223366","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}