arXiv subjects

Jani Lukkarinen

Publications and source records attributed to Jani Lukkarinen.

At least 19 recordsLinked to original sources

arXiv2026

Finite lattice kinetic equations for bosons, fermions, and discrete NLS

We introduce and study finite lattice kinetic equations for bosons, fermions, and discrete NLS. For each model this closed evolution equation provides an approximate description for the evolution of the appropriate covariance function in the system. It is obtained by truncating the cumulant hierarchy and dropping the higher order cumulants in the usual manner. To have such a reference solution should simplify controlling the full hierarchy and thus allow estimating the error from the truncation. The harmonic part is given by nearest neighbour hopping, with arbitrary symmetric interaction potential of coupling strength $λ>0$. We consider the well-posedness of the resulting evolution equation up to finite kinetic times on a finite but large enough lattice. We obtain decay of the solutions and upper bounds that are independent of $λ$ and depend on the lattice size only via some Sobolev type norms of the interaction potential and initial data. We prove that the solutions are not sensitive to how the energy conservation delta function is approximated.

math-ph↗

arXiv2025

Disorder-induced broadening of quantum momentum distribution

We study the long-time behavior of a non-interacting two-dimensional quantum gas in a weak random potential with long-range correlations. Any peaked initial momentum distribution will eventually become isotropic and broaden due to scattering events with the random potential. We derive an expression for the long-time average of the momentum distribution and test it against computer simulations. We also discuss momentum isotropization and spatial diffusion.

quant-ph↗

arXiv2025

Wasserstein error estimates between telegraph processes and Brownian motion

We provide non-asymptotic error bounds in the path Wasserstein distance with quadratic integral cost between suitable functionals of the telegraph process and the corresponding functional of Brownian motion with explicit diffusivity constant. These results cover, in particular, the well-known example of the exponential integral functional of the geometric Brownian motion. The non-asymptotic error bounds tend to zero in the so-called Kac regime. Moreover, the error bounds remain valid when the flip rate for the telegraph process is small. We assess the sharpness of the error bounds through numerical experiments.

math.PR↗

arXiv2025

Generation of chaos in the cumulant hierarchy of the stochastic Kac model

We study the time-evolution of cumulants of velocities and kinetic energies in the stochastic Kac model for velocity exchange of $N$ particles, with the aim of quantifying how fast these degrees of freedom become chaotic in a time scale in which the collision rate for each particle is order one. Chaos here is understood in the sense of the original Stoßzahlansatz, as an almost complete independence of the particle velocities which we measure by the magnitude of their cumulants up to a finite, but arbitrary order. Known spectral gap results imply that typical initial densities converge to uniform distribution on the constant energy sphere at a time which has order of $N$ expected collisions. We prove that the finite order cumulants converge to their small stationary values much faster, already at a time scale of order one collisions. The proof relies on stability analysis of the closed, but nonlinear, hierarchy of energy cumulants around the fixed point formed by their values in the stationary spherical distribution. It provides the first example of an application of the cumulant hierarchy method to control the properties of a microscopic model related to kinetic theory.

math-ph↗

arXiv2025

Smoothing Properties of a Linearization of the Three Waves Collision Operator in the bosonic Boltzmann-Nordheim Equation

We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non-equilibrium state which includes both normal fluid and a condensate. More precisely, we look at the previously postulated nonlinear Boltzmann-Nordheim equations for such systems, in a spatially homogeneous state which has an isotropic momentum distribution, and we linearize the equation around an equilibrium state which has a condensate. We study the most singular part of the linearized operator coming from the three waves collision operator for supercritical initial data. The operator has two types of singularities, one of which is similar to the marginally smoothing operator defined by the symbol $\ln(1+p^2)$. Our main result in this context is that for initial data in a certain Banach space of functions satisfying a H\"{older} type condition, at least for some finite time, evolution determined by the linearized operator improves the Hölder regularity. The main difficulty in this problem arises from the combination of a point singularity and a line singularity present in the linear operator, and we have to use certain fine-tuned function spaces in order to carry out our analysis.

math.AP↗

arXiv2023

Coagulation equations with source leading to anomalous self-similarity

We study the long-time behaviour of the solutions to Smoluchowski coagulation equations with a source term of small clusters. The source drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. The coagulation kernel is non-gelling, homogeneous, with homogeneity $γ\leq -1 $, and behaves like $x^{γ+λ} y^{-λ} $ when $y \ll x$ with $γ+2λ> 1 $. Our analysis shows that the long-time behaviour of the solutions depends on the parameters $γ$ and $λ$. More precisely, we argue that the long-time behaviour is self-similar, although the scaling of the self-similar solutions depends on the sign of $γ+λ$ and on whether $γ=-1$ or $γ< -1$. In all these cases, the scaling differs from the usual one that has been previously obtained when $γ+2λ<1$ or $γ+2λ\geq 1, γ>-1$. In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case $γ+2λ=1, γ\leq -1 $.

math-ph↗

arXiv2022

Non-power law constant flux solutions for the Smoluchowski coagulation equation

It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument.

math.AP↗

arXiv2022

Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

In this manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive non-asymptotic estimates for the corresponding time average $p$-th moments. The proof relies on coupling techniques such as coin-flip coupling, synchronous coupling and the Komlós--Major--Tusnády coupling.

math.PR↗

arXiv2022

Asymptotic localization in multicomponent mass conserving coagulation equations

In this paper we prove that the time dependent solutions of a large class of Smoluchowski coagulation equations for multicomponent systems concentrate along a particular direction of the space of cluster compositions for long times. The direction of concentration is determined by the initial distribution of clusters. These results allow to prove the uniqueness and global stability of the self-similar profile with finite mass in the case of coagulation kernels which are not identically constant, but are constant along any direction of the space of cluster compositions.

math.AP↗

arXiv2021

Localization in stationary non-equilibrium solutions for multicomponent coagulation systems

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

math-ph↗

arXiv2021

Multicomponent coagulation systems: existence and non-existence of stationary non-equilibrium solutions

We study multicomponent coagulation via the Smoluchowski coagulation equation under non-equilibrium stationary conditions induced by a source of small clusters. The coagulation kernel can be very general, merely satisfying certain power law asymptotic bounds in terms of the total number of monomers in a cluster. The bounds are characterized by two parameters and we extend previous results for one-component systems to classify the parameter values for which the above stationary solutions do or do not exist. Moreover, we also obtain criteria for the existence or non-existence of solutions which yield a constant flux of mass towards large clusters.

math.AP↗

arXiv2020

Stationary non-equilibrium solutions for coagulation systems

We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

math-ph↗

arXiv2020

Estimation of local microcanonical averages in two lattice mean-field models using coupling techniques

We consider an application of probabilistic coupling techniques which provides explicit estimates for comparison of local expectation values between label permutation invariant states, for instance, between certain microcanonical, canonical, and grand canonical ensemble expectations. A particular goal is to obtain good bounds for how such errors will decay with increasing system size. As explicit examples, we focus on two well-studied mean-field models: the discrete model of a paramagnet and the mean-field spherical model of a continuum field, both of which are related to the Curie-Weiss model. The proof is based on a construction of suitable probabilistic couplings between the relevant states, using Wasserstein fluctuation distance to control the difference between the expectations in the thermodynamic limit.

math-ph↗

arXiv2020

Pre-thermalization in a classical phonon field: slow relaxation of the number of phonons

We investigate the emergence of an astonishingly long pre-thermal plateau in a classical phonon field, here a harmonic chain with on-site pinning. Integrability is broken by a weak anharmonic on-site potential with strength $λ$. In the small $λ$ limit, the approach to equilibrium of a translation invariant initial state is described by kinetic theory. However, when the phonon band becomes narrow, we find that the (non-conserved) number of phonons relaxes on much longer time scales than kinetic. We establish rigorous bounds on the relaxation time, and develop a theory that yields exact predictions for the dissipation rate in the limit $λ\to 0$. We compare the theoretical predictions with data from molecular dynamics simulations and find good agreement. Our work shows how classical systems may exhibit phenomena which at the first glance appear to require quantization.

cond-mat.stat-mech↗

arXiv2019

Multi-state condensation in Berlin-Kac spherical models

We consider the Berlin-Kac spherical model for supercritical densities under a periodic lattice energy function which has finitely many non-degenerate global minima. Energy functions arising from nearest neighbour interactions on a rectangular lattice have a unique minimum, and in that case the supercritical fraction of the total mass condenses to the ground state of the energy function. We prove that for any sufficiently large lattice size this also happens in the case of multiple global minima, although the precise distribution of the supercritical mass and the structure of the condensate mass fluctuations may depend on the lattice size. However, in all of these cases, one can identify a bounded number of degrees of freedom forming the condensate in such a way that their fluctuations are independent from the rest of the fluid. More precisely, the original Berlin-Kac measure may be replaced by a measure where the condensate and normal fluid degrees of freedom become independent random variables, and the normal fluid part converges to the critical Gaussian free field. The proof is based on a construction of a suitable coupling between the two measures, proving that their Wasserstein distance is small enough for the error in any finite moments of the field to vanish as the lattice size is increased to infinity.

math-ph↗

arXiv2018

Random selection of factors preserves the correlation structure in a linear factor model to a high degree

In a very high-dimensional vector space, two randomly-chosen vectors are almost orthogonal with high probability. Starting from this observation, we develop a statistical factor model, the random factor model, in which factors are chosen at random based on the random projection method. Randomness of factors has the consequence that covariance matrix is well preserved in a linear factor representation. It also enables derivation of probabilistic bounds for the accuracy of the random factor representation of time-series, their cross-correlations and covariances. As an application, we analyze reproduction of time-series and their cross-correlation coefficients in the well-diversified Russell 3,000 equity index.

q-fin.ST↗

arXiv2017

Kinetic theory and thermalization of weakly interacting fermions

Weakly interacting quantum fluids allow for a natural kinetic theory description which takes into account the fermionic or bosonic nature of the interacting particles. In the simplest cases, one arrives at the Boltzmann-Nordheim equations for the reduced density matrix of the fluid. We discuss here two related topics: the kinetic theory of the fermionic Hubbard model, in which conservation of total spin results in an additional Vlasov type term in the Boltzmann equation, and the relation between kinetic theory and thermalization.

math-ph↗

arXiv2017

Kinetic theory of phonons in weakly anharmonic particle chains

The aim of this review is to develop the kinetic theory of phonons in classical particle chains to a point which allows comparing the kinetic theory of normally conducting chains, with an anharmonic pinning potential, to the kinetic theory of the anomalously conducting FPU chains. In addition to reviewing existing results from the literature, we present a streamlined derivation of the phonon Boltzmann collision operators using Wick polynomials, as well as details about the estimates which are needed to study the effect of the collision operator. This includes explicit solutions of the collisional constraints, both with and without harmonic pinning. We also recall in detail the derivation of the Green-Kubo formula for thermal conductivity in these systems, and the relation between entropy and the Boltzmann H-theorem for the phonon Boltzmann equations. The focus is in systems which are spatially translation invariant perturbations of thermal equilibrium states. We apply the results to obtain detailed predictions from kinetic theory for the Green-Kubo correlation functions, and hence the thermal conductivities, of the chain with a quartic pinning potential as well as the standard FPU-beta chain.

math-ph↗