In this paper we discuss the connections between a Vlasov–Fokker–Planck equation and an underlying microscopic particle system, and we interpret those connections in the context of the GENERIC framework (Öttinger 2005 Beyond Equilibrium Thermodynamics (New York: Wiley-Interscience)). This interpretation provides (a) a variational formulation for GENERIC systems, (b) insight into the origin of this variational formulation, and (c) an explanation of the origins of the conditions that GENERIC places on its constitutive elements, notably the so-called degeneracy or non-interaction conditions. This work shows how the general connection between large-deviation principles on one hand and gradient-flow structures on the other hand extends to non-reversible particle systems.

Institutions can provide incentives to enhance cooperation in a population where this behaviour is infrequent. This process is costly, and it is thus important to optimize the overall spending. This problem can be mathematically formulated as a multi-objective optimization problem where one wishes to minimize the cost of providing incentives while ensuring a minimum level of cooperation, sustained over time. Prior works that consider this question usually omit the stochastic effects that drive population dynamics. In this paper, we provide a rigorous analysis of this optimization problem, in a finite population and stochastic setting, studying both pairwise and multi-player cooperation dilemmas. We prove the regularity of the cost functions for providing incentives over time, characterize their asymptotic limits (infinite population size, weak selection and large selection) and show exactly when reward or punishment is more cost efficient. We show that these cost functions exhibit a phase transition phenomenon when the intensity of selection varies. By determining the critical threshold of this phase transition, we provide exact calculations for the optimal cost of the incentive, for any given intensity of selection. Numerical simulations are also provided to demonstrate analytical observations. Overall, our analysis provides for the first time a selection-dependent calculation of the optimal cost of institutional incentives (for both reward and punishment) that guarantees a minimum level of cooperation over time. It is of crucial importance for real-world applications of institutional incentives since the intensity of selection is often found to be non-extreme and specific for a given population.

Understanding the emergence of prosocial behaviours among self-interested individuals is an important problem in many scientific disciplines. Various mechanisms have been proposed to explain the evolution of such behaviours, primarily seeking the conditions under which a given mechanism can induce highest levels of cooperation. As these mechanisms usually involve costs that alter individual pay-offs, it is, however, possible that aiming for highest levels of cooperation might be detrimental for social welfare—the latter broadly defined as the total population pay-off, taking into account all costs involved for inducing increased prosocial behaviours. Herein, by comparing stochastic evolutionary models of two well-established mechanisms of prosocial behaviour—namely, peer and institutional incentives—we demonstrate that the objectives of maximizing cooperation and of maximizing social welfare are often misaligned. First, while peer punishment is often more effective than peer reward in promoting cooperation—especially with a higher impact-to-cost ratio—the opposite is true for social welfare. In fact, welfare typically decreases (increases) with this ratio for punishment (reward). Second, for institutional incentives, while maintaining similar levels of cooperation, rewards result in positive social welfare across a much broader range of parameters. Furthermore, both types of incentives often achieve optimal social welfare when their impact is moderate rather than maximal, indicating that careful planning is essential for costly institutional mechanisms to optimize social outcomes. These findings are consistent across varying mutation rates, selection intensities and game configurations. Overall, we argue for the need of adopting social welfare as the main optimization objective when designing and implementing evolutionary mechanisms for social and collective goods.

In molecular dynamics and sampling of high dimensional Gibbs measures coarse-graining is an important technique to reduce the dimensionality of the problem. We will study and quantify the coarse-graining error between the coarse-grained dynamics and an effective dynamics. The effective dynamics is a Markov process on the coarse-grained state space obtained by a closure procedure from the coarse-grained coefficients. We obtain error estimates both in relative entropy and Wasserstein distance, for both Langevin and overdamped Langevin dynamics. The approach allows for vectorial coarse-graining maps. Hereby, the quality of the chosen coarse-graining is measured by certain functional inequalities encoding the scale separation of the Gibbs measure. The method is based on error estimates between solutions of (kinetic) Fokker–Planck equations in terms of large-deviation rate functionals.

We study the Fokker–Planck equation as the many-particle limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the path-space rate functional, which characterises the large deviations from the expected trajectories, in such a way that the free energy appears explicitly. Next we use this formulation via the contraction principle to prove that the discrete time rate functional is asymptotically equivalent in the Gamma-convergence sense to the functional derived from the Wasserstein gradient discretization scheme.