Enhanced adiabatic index for hot neutron-rich matter from microscopic nuclear forces

**field**theory and find that the results are systematically larger than from typical

**mean**

**field**models. We start by constructing the finite-temperature equation of state from chiral two- and three-nucleon forces, which we then use to fit a class of extended Skyrme energy density functionals. This allows for modeling the thermal index across the full range of densities and temperatures that may be probed in simulations of core-collapse supernovae and neutron star mergers, including the low-density inhomogeneous mixed phase. For uniform matter we compare the results to analytical expressions for $\Gamma_{\mathrm{th}}$ based on Fermi liquid theory. The correlation between the thermal index and the effective masses at nuclear saturation density is studied systematically through Bayesian modeling of the nuclear equation of state. We then study the behavior of $\Gamma_{\mathrm{th}}$ in both relativistic and non-relativistic

**mean**

**field**models used in the astrophysical simulation community to complement those based on chiral effective

**field**theory constraints from our own study. We derive compact parameterization formulas for $\Gamma_{\mathrm{th}}$ across the range of densities and temperatures encountered in core collapse supernovae and binary neutron star mergers, which we suggest may be useful for the numerical simulation community.

4/10 relevant

arXiv

Combined **Mean** **Field** Limit and Non-relativistic Limit of Vlasov-Maxwell
Particle System to Vlasov-Poisson System

**mean**

**field**limit and non-relativistic limit of relativistic Vlasov-Maxwell particle system to Vlasov-Poisson equation. With the relativistic Vlasov-Maxwell particle system being a starting point, we carry out the estimates (with respect to $N$ and $c$) between the characteristic equation of both Vlasov-Maxwell particle model and Vlasov-Poisson equation, where the probabilistic method is exploited. In the last step, we take both large $N$ limit and non-relativistic limit (meaning $c$ tending to infinity) to close the argument.

10/10 relevant

arXiv

Uniform Poincar{'e} and logarithmic Sobolev inequalities for **mean** **field**
particles systems

**means**of the displacement convexity approach, or [19, 20] by Bakry-Emery technique or the recent... Expand abstract.

**mean**

**field**particles system, uniform in the number of particles, when the confinement potential have many local minimums. Our uniform log-Sobolev inequality, based on Zegarlinski's theorem for Gibbs measures, allows us to obtain the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate constant, generalizing the result of [10] by

**means**of the displacement convexity approach, or [19, 20] by Bakry-Emery technique or the recent [9] by dissipation of the Wasserstein distance.

10/10 relevant

arXiv

Fluctuations for Spatially Extended Hawkes Processes

**mean**

**field**limit u(t,x). Expand abstract.

**mean**-

**field**limit of spatially extended Hawkes processes is characterized as the unique solution u(t,x) of a neural

**field**equation (NFE). The value u(t,x) represents the membrane potential at time t of a typical neuron located in position x, embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its

**mean**

**field**limit u(t,x). Our first main result is a central limit theorem stating that the spatial distribution associated to these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural

**field**equation satisfied by u(t,x). To the best of our knowledge, this result appears to be new in the literature.

4/10 relevant

arXiv

The Dynamics of Multi-agent Multi-option Decision Making

**mean**-

**field**descriptions of disagreement and polarization. Expand abstract.

**mean**-

**field**description of the decision dynamics. Introducing the agent level back into collective decision-making models uncovers a plethora of novel collective behaviors beyond agreement and

**mean**-

**field**descriptions of disagreement and polarization. These include uniform and moderate-extremist disagreement, and switchy-and-fast versus continuous-and-slow transitions from indecision to decision. Which of these behaviors emerge depends on the model parameters, in particular, the number of agents and the number of options. Our study is grounded in Equivariant Bifurcation Theory, which allows us to formulate model-independent predictions and to develop a constructive sensitivity analysis of the decision dynamics at the organizing equivariant singularity. The localized sensitivity analysis reveals how collective decision making can be both flexible and robust in response to subtle changes in the environment or in the deciding agent interactions. Equivariant Bifurcation Theory also guides the construction of new multi-agent multi-option decision-making dynamics, for an arbitrary number of agents and an arbitrary number of options, and with fully controllable dynamical behaviors.

4/10 relevant

arXiv

Quantum fluctuations inhibit symmetry breaking in the HMF model

**mean**-

**field**theory is exact for a wide-range of classical long-range interacting systems. Is this also true once quantum fluctuations have been accounted for? As a test case we study the Hamiltonian

**Mean**

**Field**(HMF) model for a system of indistinguishable bosons which is predicted (according to

**mean**-

**field**theory) to undergo a second-order quantum phase transition at zero temperature. The ordered phase is characterized by a spontaneously broken $O(2)$ symmetry, which, despite occurring in a one-dimensional model, is not ruled out by the Mermin-Wagner theorem due to the presence of long-range interactions. Nevertheless, a spontaneously broken symmetry implies gapless Goldstone modes whose large fluctuations can restore broken symmetries. In this work, we study the influence of quantum fluctuations by projecting the Hamiltonian onto the continuous subspace of symmetry breaking

**mean**-

**field**states. We find that the energetic cost of gradients in the center of mass wavefunction inhibit the breaking of the $O(2)$ symmetry, but that the energetic cost is very small --- scaling as $\mathcal{O}(1/N^2)$. Nevertheless, for any finite $N$, no matter how large, this implies that the ground state has a restored $O(2)$ symmetry. Implications for the finite temperature phases, and classical limit, of the HMF model are discussed.

7/10 relevant

arXiv

Critical exponents and fine-grid vortex model of the dynamic vortex Mott transition in superconducting arrays

**field**with $f$ flux quantum per plaquette. The transition is induced by external driving current and thermal fluctuations near rational vortex densities set by the value of $f$, and has been observed experimentally from the scaling behavior of the differential resistivity. Recently, numerical simulations of interacting vortex models have demonstrated this behavior only near fractional $f$. A fine-grid vortex model is introduced, which allows to consider both the cases of fractional and integer $f$. The critical behavior is determined from a scaling analysis of the current-voltage relation and voltage correlations near the transition, and by Monte Carlo simulations. The critical exponents for the transition near $f=1/2$ are consistent with the experimental observations and previous numerical results from a standard vortex model. The same scaling behavior is obtained for $f=1$, in agreement with experiments. However, the estimated correlation-length exponent indicates that even at integer $f$, the critical behavior is not of

**mean**-

**field**type.

4/10 relevant

arXiv

Large Time Behaviour and the Second Eigenvalue Problem for Finite State
**Mean**-**Field** Interacting Particle Systems

**mean**-

**field**interacting particle systems with jumps. Our first main result is on the time required for convergence of the empirical measure process of the particle system to its invariant measure; we show that, there is a constant $\Lambda \geq 0$ such that, when there are $N$ particles in the system and when time is of the order $\exp\{N(\Lambda+O(1))\}$, the process has mixed well and is very close to its invariant measure. We then obtain large-$N$ asymptotics of the second eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales like $\exp\{-N\Lambda\}$. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-$N$ limit. As an application of the study of large time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain entropy function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

10/10 relevant

arXiv

Extended **Mean** **Field** Games with Singular Controls

**mean**

**field**games with continuous controls instead of singular controls and then taking approximation. Expand abstract.

**mean**

**field**games with singular controls, which arise in optimal portfolio liquidation and optimal exploitation of exhaustible resource. The interaction is through both states and controls. A relaxed solution approach is used. We prove the existence of equilibria by first considering the corresponding

**mean**

**field**games with continuous controls instead of singular controls and then taking approximation.

10/10 relevant

arXiv

Dynamical correlations and pairwise theory for the symbiotic contact process on networks

**mean**-field, even for relatively small values of vertex degree, where expressive deviations of the standard mean-

**field**are observed. Expand abstract.

**mean**-

**field**theory. This approach is compared with the ordinary one-site theory and stochastic simulations. We show that our theory significantly outperforms the one-site theory. In particular, the stationary state of the 2SCP model on random regular networks is very accurately reproduced by the pairwise

**mean**-field, even for relatively small values of vertex degree, where expressive deviations of the standard

**mean**-

**field**are observed. The pairwise approach is also able to capture the transition points accurately for heterogeneous networks and provides rich phase diagrams with transitions not predicted by the one-site method. Our theoretical results are corroborated by extensive numerical simulations.

7/10 relevant

arXiv