arXiv subjects

Giacomo Gradenigo

Publications and source records attributed to Giacomo Gradenigo.

At least 19 recordsLinked to original sources

arXiv2026

Ideal Bose-Einstein condensation in the canonical ensemble: exact asymptotic estimates from large deviations

In this work we present a large-deviations approach to the calculation of the canonical partition function for free bosons. Three-dimensional Bose-Einstein condensation is studied in the fixed-density ensemble as a function of the dimensionless density $\varrho = \rho \lambda_T^3$, with $\rho=N/L^3$ the standard particle density, $\lambda_T$ the thermal wavelength, $L$ the linear size of the box and $N$ the total number of particles. A large-deviations approach in terms of the dimensionless parameter $\ell=L/\lambda_T$ allows us to provide exact asymptotic estimates of the canonical partition function both above and below the critical density $\varrho_c$ for Bose-Einstein condensation. We show how this approach allows to explicitly account for finite-size effects and how it fully captures the first-order aspects of the transition, allowing us to explicitate its driving mechanism in terms of the competing probabilities of normal and condensed phases. The proposed large-deviations approach allows then to obtain in all regimes explicit and simple analytical expressions, at the leading order in the large parameter $\ell$, for both the average fraction of particles in the ground state, the condensate fraction $\langle n_0(\varrho) \rangle = \langle N_0(\varrho) \rangle/N$, and for its fluctuations, $\sigma_0(\varrho) = \sqrt{\langle N_0^2(\varrho)\rangle - \langle N_0(\varrho)\rangle^2}/N$, retrieving for instance the anomalous scaling $\sigma_0(\varrho)\sim 1/V^{1/3}$ in the condensed regime, $\varrho > \varrho_c$. Our large-deviations asymptotic estimate, by analytically clarifying the mixed-order nature of Bose-Einstein condensation, allows then to reveal the similarity between this transition and other mixed-order transitions, as for instance the localization transition in the Discrete Non-Linear Schr\"odinger Equation.

cond-mat.stat-mech↗

arXiv2026

Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Field Theory

Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a Hamiltonian dynamics in an intrinsic time $\tau$ which samples a microcanonical ensemble, in close analogy with the standard microcanonical approach to lattice field theory. In this contribution we present constrained symplectic quantization for relativistic quantum field theory, generalizing from the quantum mechanical case. The method is based on the analytic continuation of fields and action from $\mathbb{R}$ to $\mathbb{C}$ and on constraints that select stable intrinsic time trajectories and that simultaneously define convergent integration cycles for the microcanonical partition function. In the continuum limit we recover the Feynman generating functional with the correct real time prescription. We test the construction for a free scalar field in $1+1$ dimensions on a periodic lattice by measuring real time two point functions and by verifying Dyson Schwinger identities with the correct contact term.

hep-lat↗

arXiv2026

Constrained Symplectic Quantization II: The Free Scalar Field

Constrained symplectic quantization is a functional formulation of quantum field theory in which quantum fluctuations are sampled through a deterministic Hamiltonian flow in an auxiliary intrinsic time $\tau$. In this paper we extend the quantum-mechanical framework introduced in [1] to a relativistic scalar quantum field theory in Minkowski space-time. The construction is based on the analytic continuation of fields and action from $\mathbb{R}$ to $\mathbb{C}$ together with constraints that select stable intrinsic-time trajectories and, at the same time, define convergent integration cycles for the corresponding microcanonical functional. We show that, in the continuum limit, the microcanonical generating functional reproduces the Feynman generating functional. For the free scalar field in $1+1$ dimensions we derive the constrained equations of motion, implement the resulting dynamics numerically, and verify real-time two-point correlators, equal-time commutator relations, and Dyson--Schwinger equations including the expected contact terms.

hep-th↗

arXiv2026

Constrained Symplectic Quantization: Disclosing the Deterministic Framework Behind Quantum Mechanics

Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable $\tau$ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in 1+1 dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time 1-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.

hep-lat↗

arXiv2026

Constrained Symplectic Quantization I: the Quantum Harmonic Oscillator

Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space-time by means of a generalized microcanonical ensemble similar to the one of the standard microcanonical approach to lattice field theory. In a previous paper we showed that, for an interacting scalar field theory in 1+1-dimensions, this formalism allows to capture numerically some crucial real-time features inaccessible to any Euclidean approach to lattice field theory. Yet, the new approach was plagued by two main limitations: an ill-defined non-interacting limit and the absence of a direct formal correspondence between its correlation functions and those generated by the Feynman path integral approach. In this paper, we introduce the new \emph{"constrained symplectic quantization"} approach, for which the perfect equivalence with the Feynman path integral is proved and which is perfectly well defined for the free theory. This new approach is characterized by the analytical continuation of all fields and of the action from $\mathbb{R}$ to $\mathbb{C}$ and the presence of some constraints which guarantee the stability of the generalized Hamiltonian dynamics and the convergence of the corresponding generalized microcanonical partition function, hence the name of the theory. We show the application of this formalism to the quantum harmonic oscillator on a Minkowskian-time lattice, finding perfect agreement between one- and two-point numerical correlators and the exact quantum-mechanical results. We observe genuine real-time features such as the oscillatory propagator and the discrete excited-state energy spectrum. Our results provide strong numerical evidence that constrained symplectic quantization can sample real-time quantum-mechanical observables, offering a concrete route to overcome the limitations of Euclidean-time importance sampling.

hep-th↗

arXiv2025

A mean-field theory of effective normal modes in the Fermi-Pasta-Ulam-Tsingou model

We present a non-perturbative, mean-field theory for the Fermi-Pasta-Ulam-Tsingou model with quartic interaction, capturing the quasiperiodic features shown by the system at all energies in the thermodynamic limit. Starting from the true Hamiltonian $H$ of the system with $N$ degrees of freedom, we introduce a mean-field Hamiltonian $\mathcal{H}$ such that the difference $h_N=(H-\mathcal{H})/N$, considered as a random variable with respect to the Gibbs measure, tends to zero as $N\to\infty$, in probabilistic sense. The dynamics of the mean-field Hamiltonian $\mathcal{H}$ consists of $N$ independent oscillation modes with renormalized frequencies $\Omega_k = \omega_k\sqrt{1+\gamma(\varepsilon)}$, $\omega_k$ being the frequency of the $k$-th normal mode of the linearized system, whereas $\gamma(\varepsilon)$ is an explicit function of the specific energy $\varepsilon$ of the system. Analytical predictions drawn from the effective Langevin equations ruling the dynamics of such oscillation modes are successfully compared with the numerical data from the original Hamiltonian dynamics. Such a simple decomposition of the true dynamics into $N$ effective normal modes holds at all energy scales, i.e. from the quasi-integrable regime to the strongly chaotic one.

cond-mat.stat-mech↗

arXiv2025

Entropy production and irreversibility in the linearized stochastic Amari neural model

One among the most intriguing results coming from the application of statistical mechanics to the study of brain is the understanding that it, as a dynamical system, is inherently out of equilibrium. In the realm of non-equilibrium statistical mechanics and stochastic processes the standard observable computed to discriminate whether a system is at equilibrium or not is the entropy produced along the dynamics. For this reason we present here a detailed calculation of the entropy production in the Amari model, a coarse-grained model of the brain neural network, consisting in an integro-differential equation for the neural activity field, when stochasticity is added to the original dynamics. Since the way to add stochasticity is always to some extent arbitrary, i.e., in particular for coarse-grained models, there is no general prescription to do it, we precisely investigate the interplay between the noise properties and the original model features, discussing in which cases the stationary state is of thermal equilibrium and which cases is out of equilibrium, providing explicit and simple formulas. We also show how, following for the derivation the particular case considered, how the entropy production rate is related to the variation in time of the Shannon entropy of the system.

cond-mat.dis-nn↗

arXiv2025

Generalized Lotka-Volterra model with sparse interactions: non-Gaussian effects and topological multiple-equilibria phase

We study the equilibrium phases of a generalized Lotka-Volterra model characterized by a species interaction matrix which is random, sparse and symmetric. Dynamical fluctuations are modeled by a demographic noise with amplitude proportional to the effective temperature T. The equilibrium distribution of species abundances is obtained by means of the cavity method and the Belief Propagation equations, which allow for an exact solution on sparse networks. Our results reveal a rich and non-trivial phenomenology that deviates significantly from the predictions of fully connected models. Consistently with data from real ecosystems, which are characterized by sparse rather than dense interaction networks, we find strong deviations from Gaussianity in the distribution of abundances. In addition to the study of these deviations from Gaussianity, which are not related to multiple-equilibria, we also identified a novel topological glass phase, present at both finite temperature, as shown here, and at T=0, as previously suggested in the literature. The peculiarity of this phase, which differs from the multiple-equilibria phase of fully-connected networks, is its strong dependence on the presence of extinctions. These findings provide new insights into how network topology and disorder influence ecological networks, particularly emphasizing that sparsity is a crucial feature for accurately modeling real-world ecological phenomena.

cond-mat.stat-mech↗

arXiv2025

Localization and "classical entanglement'' in the Discrete Non-Linear Schr\"odinger Equation

We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the Discrete Non-Linear Schr\"odinger Equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called {\it ``classical entanglement''}. Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called $S_{\mathrm{ent}}$, grows with the system size $N$ in the localized phase as $S_{\mathrm{ent}}(N) \sim \log(N)$, therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of {\it ``classical entanglement''} beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations.

cond-mat.stat-mech↗

arXiv2024

Symplectic Quantization and Minkowskian Statistical Mechanics: simulations on a 1+1 lattice

We introduce symplectic quantization, a novel functional approach to quantum field theory which allows to sample quantum fields fluctuations directly in Minkowski space-time, at variance with the traditional importance sampling protocols, well defined only for Euclidean Field Theory. This importance sampling procedure is realized by means of a deterministic dynamics generated by Hamilton-like equations evolving with respect to an auxiliary time parameter $\tau$. In this framework, expectation values over quantum fluctuations are computed as dynamical averages along the trajectories parameterized by $\tau$. Assuming ergodicity, this is equivalent to sample a microcanonical partition function. Then, by means of a large-M calculation, where M is the number of degrees of freedom on the lattice, we show that the microcanonical correlation functions are equivalent to those generated by a Minkowskian canonical theory where quantum fields fluctuations are weighted by the factor $\exp(S/\hbar )$, with $S$ being the original relativistic action of the system.

hep-lat↗

arXiv2024

On the foundations of statistical mechanics

Although not as wide, and popular, as that of quantum mechanics, the investigation of fundamental aspects of statistical mechanics constitutes an important research field in the building of modern physics. Besides the interest for itself, both for physicists and philosophers, and the obvious pedagogical motivations, there is a further, compelling reason for a thorough understanding of the subject. The fast development of models and methods at the edge of the established domain of the field requires indeed a deep reflection on the essential aspects of the theory, which are at the basis of its success. These elements should never be disregarded when trying to expand the domain of statistical mechanics to systems with novel, little known features. It is thus important to (re)consider in a careful way the main ingredients involved in the foundations of statistical mechanics. Among those, a primary role is covered by the dynamical aspects (e.g. presence of chaos), the emergence of collective features for large systems, and the use of probability in the building of a consistent statistical description of physical systems. With this goal in mind, in the present review we aim at providing a consistent picture of the state of the art of the subject, both in the classical and in the quantum realm. In particular, we will highlight the similarities of the key technical and conceptual steps with emphasis on the relevance of the many degrees of freedom, to justify the use of statistical ensembles in the two domains.

cond-mat.stat-mech↗

arXiv2024

Symplectic Quantization: numerical results for the Feynman propagator on a 1+1 lattice and the theoretical relation with Quantum Field Theory

We present here the first lattice simulation of symplectic quantization, a new functional approach to quantum field theory which allows to define an algorithm to numerically sample the quantum fluctuations of fields directly in Minkowski space-time, at variance with all other present approaches. Symplectic quantization is characterized by a Hamiltonian deterministic dynamics evolving with respect to an additional time parameter $\tau$ analogous to the fictious time of stochastic quantization. The difference between stochastic quantization and the present approach is that the former is well defined only for Euclidean field theories, while the latter allows to sample the causal structure of space-time. In this work we present the numerical study of a real scalar field theory on a 1+1 space-time lattice with a $\lambda\phi^4$ interaction. We find that for $\lambda\ll1$ the two-point correlation function obtained numerically reproduces qualitatively well the shape of the free Feynman propagator. Within symplectic quantization the expectation values over quantum fluctuations are computed as dynamical averages along the dynamics in $\tau$, in force of a natural ergodic hypothesis connecting Hamiltonian dynamics with a generalized microcanonical ensemble. Analytically, we prove that this microcanonical ensemble, in the continuum limit, is equivalent to a canonical-like one where the probability density of field configurations is $P[\phi]\propto\exp(zS[\phi]/\hbar)$. The results from our simulations correspond to the value $z=1$ of the parameter in the canonical weight, which in this case is a well-defined probability density for field configurations in causal space-time, provided that a lower bounded interaction potential is considered. The form proposed for $P[\phi]$ suggests that our theory can be connected to ordinary quantum field theory by analytic continuation in the complex-$z$ plane.

hep-lat↗

arXiv2024

Probing marginal stability in the spherical $p=2$ model

In this paper we investigate the marginally stable nature of the low-temperature trivial spin glass phase in the spherical $p=2$ spin glass, by perturbing the system with three different kinds of non-linear interactions. In particular, we compare the effect of three additional dense four-body interactions: ferromagnetic couplings, purely disordered couplings and couplings with competing disordered and ferromagnetic interactions. Our study, characterized by the effort to present in a clear and pedagogical way the derivation of all the results, shows that the marginal stability property of the spherical spin glass depends in fact on which kind of perturbation is applied to the system: in general, a certain degree of frustration is needed also in the additional terms in order to induce a transition from a trivial to a non-trivial spin-glass phase. On the contrary, the addition of generic non-frustrated interactions does not destabilize the trivial spin-glass phase.

cond-mat.dis-nn↗

arXiv2024

Symplectic quantization III: Non-relativistic limit

First of all we shortly illustrate how the symplectic quantization scheme [Gradenigo and Livi, 2021] can be applied to a relativistic field theory with self-interaction. Taking inspiration from the stochastic quantization method by Parisi and Wu, this procedure is based on considering explicitly the role of an intrinsic time variable, associated with quantum fluctuations. The major part of this paper is devoted to showing how the symplectic quantization scheme can be extended to the non-relativistic limit for a Schr\"odinger-like field. Then we also discuss how one can obtain from this non-relativistic theory a linear Schr\"odinger equation for the single-particle wavefunction. This further passage is based on a suitable coarse-graining procedure, when self-interaction terms can be neglected, with respect to interactions with any external field. In the Appendix we complete our survey on symplectic quantization by discussing how this scheme applies to a non-relativistic particle under the action of a generic external potential.

cond-mat.stat-mech↗

arXiv2023

Universality class of the mode-locked glassy random laser

By means of enhanced Monte Carlo numerical simulations parallelized on GPU's we study the critical properties of the spin-glass-like model for the mode-locked glassy random laser, a $4$-spin model with complex spins with a global spherical constraint and quenched random interactions. Using two different boundary conditions for the mode frequencies we identify the critical points and the critical indices of the random lasing phase transition using , with finite size scaling techniques. The outcome of the scaling analysis is that the mode-locked random laser is in a mean-field universality class, though different from the mean-field class of the Random Energy Model and the glassy random laser in the narrow band approximation, that is, the fully connected version of the present model. The low temperature (high pumping) phase is finally characterized by means of the overlap distribution and evidence for the onset of replica symmetry breaking in the lasing regime is provided.

cond-mat.dis-nn↗

arXiv2022

Intensity pseudo-localized phase in the glassy random laser

Evidence of an emergent pseudo-localized phase characterizing the low-temperature replica symmetry breaking phase of the complex disordered models for glassy light is provided in the mode-locked random laser model. A pseudo-localized phase corresponds to a state in which the intensity of light modes is neither equipartited among all modes nor really localized on few of them. Such a hybrid phase has been recently characterized in other models, such as the Discrete Non-Linear Schr\"odinger equation, just as a finite size effect, while in the low temperature phase of the glassy random laser it seems to be robust in the limit of large size.

cond-mat.dis-nn↗

arXiv2021

Thermalization without chaos in harmonic systems

Recent numerical results showed that thermalization of Fourier modes is achieved in short time-scales in the Toda model, despite its integrability and the absence of chaos. Here we provide numerical evidence that the scenario according to which chaos is irrelevant for thermalization is realized even in the simplest of all classical integrable system: the harmonic chain. We study relaxation from an atypical condition given with respect to "random" modes, showing that a thermal state with equilibrium properties is attained in short times. Such a result is independent from the orthonormal base used to represent the chain state, provided it is random.

cond-mat.stat-mech↗

arXiv2021

First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension

We consider a single run-and-tumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zero-mean Gaussian distribution and that the run times are exponentially distributed. We investigate the probability distribution $P(X,N)$ of the position $X$ of the particle after $N$ runs, with $N\gg 1$. We show that in the regime $ X \sim N^{3/4}$ the distribution $P(X,N)$ has a large deviation form with a rate function characterized by a discontinuous derivative at the critical value $X=X_c>0$. The same is true for $X=-X_c$ due to the symmetry of $P(X,N)$. We show that this singularity corresponds to a first-order condensation transition: for $X>X_c$ a single large jump dominates the RTP trajectory. We consider the participation ratio of the single-run displacements as the order parameter of the system, showing that this quantity is discontinuous at $X=X_c$. Our results are supported by numerical simulations performed with a constrained Markov chain Monte Carlo algorithm.

cond-mat.stat-mech↗