Monday, 28 September 2026
TBA
Han et al. [PRL 137102 (2024)] have recently introduced a classical stochastic lattice gas which, in addition to particle conservation, also conserves the particles’ dipole moment. The dipole-conserving gas (DCG) exhibits unusual macroscopic scaling behaviors, different from those of lattice gases that conserve only the number of particles. We investigated some basic relaxation and fluctuation properties of this model. Using MFT, we determined the probability distribution of, and the optimal path of the gas leading to, a large deviation in the form of a macroscopic void of a given size in an initially uniform DCG at equilibrium. We also calculated the variance of nonequilibrium fluctuations of current when starting from an (either deterministic, or random) constant density at t = 0.
The celebrated Arrhenius law (AL) captures the activation time of a single diffusing particle from a metastable state, yielding an exponential dependence on the activation barrier. We use Macroscopic Fluctuation Theory to extend this law to many diffusing and interacting particles. Our analysis reveals a new universality class of activation, but also retains the Arrhenius universality depending on the nature of the inter-particle interactions.
We first introduce a probabilistic representation of measures described by the Matrix Product Ansatz. By suitably enlarging the state space, we construct a coupling measure which can be represented as a Markov bridge on the enlarged space. As a consequence, the original MPA measure can also be expressed as a mixture of inhomogeneous product measures, with mixing law given by a Markov bridge. We then exploit the Markovian structure on the enlarged state space to study large deviations for MPA measures. Finally, we discuss the infinite-dimensional case through the boundary-driven TASEP, recovering the known large deviation functional for its stationary density profile. Joint work with Davide Gabrielli
I will discuss two settings where the variational problem within MFT can be solved to yield explicit results on nonequilibrium fluctuations. The first concerns tracer and current statistics in a single-file gas of Brownian hard rods. Exploiting a duality with a point-particle system, we exactly compute the cumulant-generating functions and optimal fluctuation paths for two domain-wall initial ensembles. Our results reveal an intriguing connection with the height field of the dual point-particle system, linking single-file transport to interface dynamics. The second concerns steady-state density large deviations in boundary-driven diffusive systems with a bulk drive. Using an auxiliary-field transformation, we obtain perturbative results for generic systems, including in arbitrary dimensions. Our formulation reveals the dynamical origin of long-range correlations in the steady state and how the onset of non-locality depends on the bulk drive.
Over the past two decades, the Macroscopic Fluctuation Theory (MFT) has provided a universal framework to deal with classical diffusive non-equilibrium systems. A natural question is whether this framework can be extended to quantum mechanics to capture the statistics of inherently quantum phenomena, such as interference and entanglement, in diffusive, out-of-equilibrium systems. In this talk, we first review key insights from model systems, including Quantum Exclusion Processes. We then present recent progress toward formulating a Quantum Mesoscopic Fluctuation Theory (QMFT) and conclude by outlining the open challenges and steps required to achieve a comprehensive theory.
Tuesday, 29 September 2026
This pedagogical lecture will discuss a discrete formulation of macroscopic fluctuation theory (arising from the large-spin limit of interacting particle systems) and duality properties.
In the first part, I will introduce Markov duality and explain how it provides an effective tool for studying non-equilibrium stochastic systems. The harmonic process—an integrable model of heat conduction—will serve as the main example. I will review its algebraic structure and show how duality gives access to correlation functions and to an explicit characterization of the non-equilibrium stationary state. This exact microscopic information makes it possible to verify some predictions of macroscopic fluctuation theory, while suggesting the need for extensions to include singular profiles.
In the second part, I will discuss dynamical large deviations in the large-spin regime. For a large class of models, I will explain how a path-space large-deviation principle leads to a finite-dimensional Hamiltonian theory on the lattice—a discrete counterpart of the usual macroscopic fluctuation theory. The associated Hamilton equations and variational principles describe the optimal trajectories responsible for rare density and current fluctuations. I will conclude by discussing the connections among discrete and continuum MFT and fluctuation symmetries.
We consider exclusion processes as models for active matter systems, which undergo motility-induced phase separation [1] and as well dynamical pattern-forming behaviour [2]. I will present examples of phase transitions in these systems, as well as discussing the mechanisms for (rare) transitions between metastable states.
[1] Kourbane-Houssene, Erignoux, Bodineau and Tailleur, PRL 120, 268003 (2018).
[2] Mason, Jack, and Bruna, Nat Comms 16, 6017 (2025).
In active systems detailed balance is broken at the level of each individual constituent. In this talk I will show that, in several cases, the probability of nucleating the stable state from a metastable one can be computed analytically although the instanton is not the time-reversal of the relaxation dynamics. We will furthermore discuss that their stationary measure can be generically expected to be non-local whenever at least one conservation law is present, and that this property has a surprising relation with nucleation probabilities. Our results will be mostly based on the weak-noise regime of a number of field theories that were proposed to describe active systems in the past.
“Designs” are sets of points, for example on a sphere, that have the property that averages of polynomials over the set coincide with averages over the whole sphere. Similarly for the unitary group. One can treat these points as interacting particles, and use liquid theory to study the problem. Presented from a physicist's point of view.
Wednesday, 30 September 2026
Macroscopic Fluctuation Theory (MFT) describes non-equilibrium large deviations through non-linear coupled PDEs called the MFT equations. Recent works, including the one by Mallick, Moriya and the lecturer[1], showed that these MFT equations for the symmetric simple exclusion process (SEP) and a few other models can be mapped to classical integrable systems and can be solved via the Inverse Scattering Methods (ISM).
In this two-hour pedagogical lecture, we provide an introduction to inverse scattering methods applied to MFT. We begin with the conventional continuous MFT for systems such as the SEP, illustrating how the underlying variational equations can be mapped to a classical integrable system called the AKNS system and can be solved through Lax pairs and spectral theory. We then extend these techniques to discrete lattice systems.
References:
[1] K. Mallick, H. Moriya, T. Sasamoto, Phys. Rev. Lett. 118, 160601 (2022).
[2] K. Mallick, H. Moriya, T. Sasamoto, JSTAT {¥bf 2024} 074001 (2024).
We show the relationship between the strongly non-linear limit (also termed the dispersionless or the Whitham limit) of the macroscopic fluctuation theory of certain statistical models and the inverse scattering method. We show that in the strongly non-linear limit the inverse scattering problem can be solved using the steepest descent method of the associated Riemann–Hilbert problem. The form of the strongly-nonlinear theory serves as a formal classification scheme for the different models studied using large fluctuation theory that exhibit integrability.
We consider a system of N particles interacting via stochastic binary collisions, like the celebrated Kac's model. We discuss the large deviations asymptotic of the empirical measure and flow in a fixed time interval, reviewing some relevant results and open problems. As an application, we discuss the probability of fluctuations of the total number of collisions in the joint limit in which both the number of particles and the time interval diverge.
TBA
Thursday, 01 October 2026
TBA
We consider a system of harmonic oscillators in contact with reservoirs, described by a bosonic Ornstein–Uhlenbeck semigroup. We show that the Glauber–Sudarshan normal-order map is a contraction between suitable Gaussian Hilbert spaces, and that it intertwines the quantum dynamics with a classical Ornstein–Uhlenbeck process on phase space, with the same drift and a noise corrected by a zero-point term. As applications, we derive the sectoriality and the spectrum of the bosonic Ornstein–Uhlenbeck generator. Joint work with L. Bertini, G. Posta and C. Presilla
While Macroscopic Fluctuation Theory (MFT) has been highly successful in analyzing non-equilibrium steady states, its application to non-steady-state processes remains limited. In this talk, we will present our MFT analysis of the relaxation process of one-dimensional boundary-driven diffusive systems coupled to particle reservoirs at both ends. We derive the MFT action for the integrated current and obtain the corresponding MFT equations, together with their initial and boundary conditions. Using this formulation, we exactly derive the current variance for systems with a constant diffusion coefficient and arbitrary mobility, and the cumulant generating function for the current in Reflective Brownian Motion (RBM). Our results demonstrate that non-steady current fluctuations during the approach to a steady state can be quantitatively described within the MFT framework.
This talk is based on joint work with Tomohiro Sasamoto (arXiv:2605.27275).
A measure of similarity or overlap between different configurations or individuals appears in many complex systems such as spin glasses, evolving populations, optimization problems and neural networks. In the context of spin glasses, the concept was introduced by Edwards and Anderson 50 years ago and gives a measure of similarity between pairs of spin configurations. In the context of evolving populations, the overlap measures the similarity between genomes of individuals. In general the overlaps fluctuate and have non-trivial statistics. Giorgio Parisi's replica theory of mean field spin glasses predicted a universal form for the overlap statistics. In this talk I will compare these predictions with the statistics for different models of disordered systems and for evolving populations.
Monday, 05 October 2026
I these lectures I will pedagogically cover the Ballistic Macroscopic Fluctuation Theory, a hydrodynamic fluctuation theory for systems admitting ballistic transport. As time permits, I will explain: (1) its basic physical principles, (2) its action formulation, which adapts the Macroscopic Fluctuation Theory to ballistic transport and gives equations for the large-deviation theory of currents, (3) its observable-based formulation, which connects with fluctuating hydrodynamics and gives predictions for Euler-scale correlation functions, (4) its connection with the Ballistic Fluctuation theory, which provides an alternative framework for the large-deviation theory of currents. Throughout, I will try to show the examples of many-body integrable systems concentrating on the hard rods for simplicity (as linearly degenerate systems without shocks), and TASEP (as truly non-linear systems admitting shocks). This is based on various works including with Gabriele Perfetto, Tomohiro Sasamoto and Takato Yoshimura, and with Jason Myers, M. J. Bhaseen and Rosemary J. Harris.
Nonlinear fluctuating hydrodynamics (NLFHD) is a versatile hydrodynamic theory for describing fluctuations at typical scales. While the theory is generally expected to apply only to non-integrable systems, in this talk I will argue that its underlying ideas can be extended to integrable systems, where they correctly capture typical fluctuations. I will then use this extended framework to derive an effective action for the fluctuations of a single normal mode by integrating out the remaining modes. I will show that, at the level of two-point functions, the resulting dynamics is governed by the Edwards–Wilkinson equation.
Genetic drift plays a crucial role in evolution by enhancing diversity within ecosystems. On the other hand, active self-propulsion can drive systems into phase separated regions promoting homogeneity within a spatial region. To analyze the intricate effects produced by such competing processes, we introduce a minimal model for a proliferating active population, where run-and-tumble migration is combined with a stochastic birth-death process. Beginning with the microscopic dynamics of the particles, we derive the corresponding hydrodynamic equations and validate them through numerical simulations. We demonstrate that the inclusion of activity leads to the emergence of novel morphological patterns, which are sensitive to the specific rules governing migration. Furthermore, we show that activity significantly alters the characteristics of invasion waves, increasing their propagation speed. However, beyond a threshold, excessive activity disrupts the wave-like nature of the invasion process. We also extend our study to two dimensions and analyze the instabilities that lead to pattern formation in such systems.
Genetic drift plays a crucial role in evolution by enhancing diversity within ecosystems. On the other hand, active self-propulsion can drive systems into phase separated regions promoting homogeneity within a spatial region. To analyze the intricate effects produced by such competing processes, we introduce a minimal model for a proliferating active population, where run-and-tumble migration is combined with a stochastic birth-death process. Beginning with the microscopic dynamics of the particles, we derive the corresponding hydrodynamic equations and validate them through numerical simulations. We demonstrate that the inclusion of activity leads to the emergence of novel morphological patterns, which are sensitive to the specific rules governing migration. Furthermore, we show that activity significantly alters the characteristics of invasion waves, increasing their propagation speed. However, beyond a threshold, excessive activity disrupts the wave-like nature of the invasion process. We also extend our study to two dimensions and analyze the instabilities that lead to pattern formation in such systems.
Tuesday, 06 October 2026
(Joint works with Gideon Amir, Ofer Busani and Ellen Saada)
Abstract: We study invariant measures and hydrodynamic limit for multilane asymmetric exclusion process.
The invariant measures problem is intermediate between well understood 1d and far less understood multi-d.
We identify a domain of parameters where a full characterization can be obtained including 2D like blocking measures.
We then study lane by lane hydrodynamic limit under a double horizontal-vertical scalling and prove that the limit obtained coïncides with
the limit of a hyperbolic system of conservation laws with relaxation. In the two-lane case we obtain a complete two-parameter phase diagram of
the current-density relation illustrating the competition between lanes.
In a joint work with Assaf Shapira and Federico Sau, we derive the Euler (hyperbolic) hydrodynamic limit for the directed exclusion process (DEP), a one-dimensional conservative interacting particle system that preserves particle-hole symmetry while breaking left-right symmetry. The proof relies on an explicit multi-process coupling which guarantees a strong form of attractiveness and macroscopic stability for the particle system.
Can a deterministic evolution remain random when its microscopic uncertainty vanishes? Turbulence offers a striking setting for this question: as viscosity tends to zero, infinitesimal perturbations can produce finite differences in the flow within a finite time. This phenomenon, known as spontaneous stochasticity, goes beyond the familiar sensitivity to initial conditions of deterministic chaos. I will introduce this perspective through the interplay between singular dynamics, persistent energy dissipation, and loss of predictability. Using examples ranging from an unstable shear layer to three-dimensional turbulence, I will discuss numerical evidence that uncertainty spreads from small to large scales through a statistically self-similar process, largely independent of how the perturbations are introduced. Particular attention will be paid to universality and to the anomalous scaling of error growth associated with intermittent fluctuations.
We consider systems of interacting particles described by an underdamped (second order) Langevin equation. We introduce an associated equation of fluctuating hydrodynamics, which can be interpreted as a stochastic version of a Vlasov-Fokker-Planck equation. We show that a dichotomy previously known for purely diffusive systems holds in this setting as well: solutions exist only for suitable atomic initial data, but not for smooth initial data. The class of systems covered includes several models of active matter. We will then sketch how first order fluctuating hydrodynamics can be used to learn the associated evolution operator for the hydrodynamic limit in thermodynamic form, with rigorous error estimates for the symmetric exclusion process.
In this talk, we consider the class of Harmonic models on a 1D lattice with open boundaries, for which large amounts of mass can be transferred in a single jump.
We show that the non-equilibrium steady state can be explicitly characterised via a mixture of inhomogeneous product measures, with mixing parameters given by the order statistics of uniform random variables. This structure allows to compute the macroscopic large deviation function and to establish an additivity principle.
Furthermore, we analyse the steady-state Shannon entropy. We show that the leading O(N) term coincides with the one of local equilibrium, while the first O(1) correction depends only on the rescaled two-point truncated correlation and matches the one of Brownian bridge as N→∞.
What dynamics emerge when a stochastic process is conditioned on a rare event?
I will show how Doob transforms and level-2.5 large deviations use empirical densities and currents to characterize the resulting effective processes.
This variational viewpoint leads to the Schrödinger problem of prescribed endpoint laws, and to its richer path-space formulation for interacting particles.
Wednesday, 07 October 2026
We consider a finite one-dimensional unpinned harmonic chain with stochastic momentum exchange and Langevin heat baths at its endpoints, possibly at different temperatures. The bulk dynamics has three locally conserved quantities: energy, momentum and stretch, and exhibits superdiffusive energy transport on the time scale $n^{3/2}$, where $n$ is the size of the system. We prove the convergence of the averaged energy profile on this superdiffusive scale and identify the thermal boundary conditions that emerge from the microscopic dynamics. The limiting evolution is governed by the $3/4$-fractional Neumann Laplacian supplemented by explicit non-local boundary terms. In contrast with the usual Dirichlet boundary conditions of diffusive systems, the coupling to the thermostats produces a non-local boundary layer, encoding the absorption, reflection and transmission of long-wavelength phonons at the boundaries. This yields a new type of boundary condition for fractional diffusion and provides its rigorous derivation from an underlying microscopic dynamics with local interactions. This is joint work with Stefano Olla.
We introduce a novel exclusion process with two conservation laws, mass and energy, designed to mimic continuous systems like interacting oscillators. Unlike conventional multi-species processes where only particle numbers are conserved, our model exhibits a wide variety of universality classes under nonlinear fluctuating hydrodynamics (NFH) depending on parameter choices. In this talk, we study the stationary fluctuations of these conserved quantities. We rigorously demonstrate that, under a suitable choice of parameters, the fluctuation fields converge to uncoupled stochastic Burgers equations (SBE) in the scaling limit. In addition, we present a general and rigorous proof of the diagonalizability of the macroscopic current's Jacobian matrix with distinct real eigenvalues. This mathematical justification addresses a property often assumed without proof in the physics literature, offering a firm foundation for a broad class of multi-component systems. This talk is based on a joint work with Hugo Da Cunha.
We derive the structure of macroscopic dynamical fluctuations for nonequilibrium systems satisfying local detailed balance. From that follows the structure of relaxation equations to steady nonequilibria. This approach does not start from more microscopic interacting particle systems but immediately gives the form of thermodynamically consistent relaxation, much in the spirit of Onsager's project. We give examples, including a modification of the Lorenz model to a recharge-discharge oscillator (killing its butterfly).
This is an introductory talk on key features of Generalized Hydrodynamics. In particularly, covered are generalized Gibbs ensembles including their large deviations and the role of the time-dependent random Lax matrix.
Thursday, 08 October 2026
Large deviation theory provides a common theoretical framework to compute probabilities of rare events in stochastic systems out of equilibrium. The theory consists of a saddlepoint evaluation of the path integral that describes the stochastic process under study, and has successfully been used in various physical systems such as interface growth, active matter, lattice gases and the macroscopic fluctuation theory, fluid dynamics and turbulence, and so forth.
In this talk, I will describe recent progress in going beyond leading-order large deviation asymptotics, developing tractable and general methods to exactly evaluate 1-loop or Gaussian corrections around nontrivial large deviation minimizers for weak noise Langevin equations and field theories. To compute the corresponding large deviation prefactors, on the one hand, I will introduce an approach based on matrix Riccati differential equations, and on the other hand, I will show how formulating the prefactor in terms of Fredholm determinants or renormalized Carleman-Fredholm determinants and operator traces makes it feasible to numerically evaluate these corrections in very high-dimensional systems.
To illustrate these points, I will show multiple examples of precise rare event estimates in statistical field theories, such as extreme growth events in the one-dimensional KPZ equation at short times, extreme concentrations of a randomly advected passive scalar, and extreme vortices and strain events in the stochastically forced incompressible Navier-Stokes equations.
Rare fluctuations provide a powerful way to probe phase transitions and non-equilibrium phenomena, but their exponentially small probabilities make them challenging to access numerically. In this talk, I will discuss a general importance-sampling strategy for sampling rare events, and illustrate how the same underlying idea can be applied both to equilibrium configurations and to dynamical trajectories. I will first consider equilibrium free-energy landscapes, where umbrella sampling can be used to probe atypical values of an order parameter and reconstruct its large-deviation rate function, illustrated in the context of the thermodynamics of three-dimensional supercooled liquids and the liquid-to-glass transition. I will then turn to dynamical large deviations, encoding trajectory stochasticity into random numbers so that importance sampling can be performed directly in trajectory space, applied to a one-dimensional gas of Brownian hard rods (tracer displacement and integrated particle current), compared with recent exact MFT predictions.
Irreversible aggregation is a paradigmatic nonequilibrium process, with the product-kernel serving as a canonical model for finite-time gelation. While its typical mean-field behavior is well understood, an exact theory for its rare fluctuations has remained elusive. In this talk, I will present an exact fluctuation theory for the product-kernel using a canonical path-integral formulation. By exactly solving the associated Euler-Lagrange equations, we determine the explicit instanton trajectories, revealing a dynamical phase transition where a macroscopic gel emerges as a time-dependent background potential. This framework yields the complete, non-convex LDF and the exact phase diagram. We demonstrate ensemble inequivalence between canonical and grand-canonical descriptions. Unlike equilibrium systems where inequivalence stems from non-additive Hamiltonians, here it arises purely from the global, dynamical coupling inherent in the collision rates.
We show that, on a d-dimensional hypercubic lattice with d>1, conserved-mass transport processes with multidirectional hopping generically exhibit power-law correlations. The underlying mechanism is multidirectional hopping, where several chunks of mass, or multiple particles, can hop simultaneously and in a coordinated manner from a lattice site along different directions, leading to a violation of detailed balance. We illustrate this mechanism through two important classes of dynamics: (i) center-of-mass-conserving dynamics and (ii) chiral dynamics, both of which produce algebraically decaying correlations. These findings demonstrate how coordinated multidirectional transport can provide a generic microscopic mechanism for generating power-law correlations in nonequilibrium conserved systems.
We study the current fluctuations of the one-dimensional symmetric exclusion process evolving under a quenched initial condition. The macroscopic fluctuation theory is employed to probe the fluctuations and it is subsequently reformulated in terms of the potential in the inverse scattering method, leading to the boundary equation that determines the final profile of the potential. Then, we solve this nonlinear functional integral equation by applying the perturbation method up to the sufficient order to compute the fourth cumulant.
Friday, 09 October 2026
We study density and current fluctuations in one-dimensional systems of run-and-tumble particles, with and without interactions. Starting from Dean–Kawasaki equations, we derive density correlation functions and characterize the fluctuations of the time-integrated current, revealing distinct long-time behaviors for short- and long-range interactions. We then develop a macroscopic fluctuation theory that retains the full Poissonian statistics of tumbling events and gives access to current large deviations. For non-interacting particles, this framework recovers known results and captures the crossover between short-time ballistic and long-time diffusive regimes.