Many-particle Systems Research Articles

Derivation of a fractional cross-diffusion system as the limit of a stochastic many-particle system driven by Lévy noise

In this article a fractional cross-diffusion system is derived as the rigorous many-particle limit of a multi-species system of moderately interacting particles that is driven by Lévy noise. The form of the mutual interaction is motivated by the porous medium equation with fractional potential pressure. Our approach is based on the techniques developed by Oelschläger (1989) and Stevens (2000), in the latter of which the convergence of a regularization of the empirical measure to the solution of a correspondingly regularized macroscopic system is shown. A well-posedness result and the non-negativity of solutions are proved for the regularized macroscopic system, which then yields the same results for the non-regularized fractional cross-diffusion system in the limit.

Applying the symmetry groups to study the n body problem

We introduce an algebraic method to study local stability in the Newtonian n-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. Using representation theory of finite groups, we explicitly found the eigenvalues of certain 8×8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.

Breakdown of quantum-classical correspondence and dynamical generation of entanglement

The {\it exchange} interaction arising from the particle indistinguishability is of central importance to physics of many-particle quantum systems. Here we study analytically the dynamical generation of quantum entanglement induced by this interaction in an isolated system, namely, an ideal Fermi gas confined in a chaotic cavity, which evolves unitarily from a non-Gaussian pure state. We find that the breakdown of the quantum-classical correspondence of particle motion, via dramatically changing the spatial structure of many-body wavefunction, leads to profound changes of the entanglement structure. Furthermore, for a class of initial states, such change leads to the approach to thermal equilibrium everywhere in the cavity, with the well-known Ehrenfest time in quantum chaos as the thermalization time. Specifically, the quantum expectation values of various correlation functions at different spatial scales are all determined by the Fermi-Dirac distribution. In addition, by using the reduced density matrix (RDM) and the entanglement entropy (EE) as local probes, we find that the gas inside a subsystem is at equilibrium with that outside, and its thermal entropy is the EE, even though the whole system is in a pure state. As a by-product of this work, we provide an analytical solution supporting an important conjecture on thermalization, made and numerically studied by Garrison and Grover in: Phys. Rev. X \textbf{8}, 021026 (2018), and strengthen its statement.

Classical and relativistic n-body problem: from Levi-Civita to the most advanced interplanetary missions

The n-body problem is one of the most important issue in Celestial Mechanics. This article aims to retrace the historical and scientific events that led the Paduan mathematician, Tullio Levi-Civita, to deal with the problem first from a classic and then a relativistic point of view. We describe Levi-Civita's contributions to the theory of relativity focusing on his epistolary exchanges with Einstein, on the problem of secular acceleration and on the proof of Brillouin's cancellation principle. We also point out that the themes treated by Levi-Civita are very topical. Specifically, we analyse how the mathematical formalism used nowadays to test General Relativity can be found in Levi-Civita's texts and evolves over the years up to the current Parametrized version of the Post-Newtonian approximation (PPN) which is used in high precision contexts such as important space missions designed also to test General Relativity and which aim to estimate with very high accuracy the PPN parameters.

Wavefunction structure in quantum many-fermion systems with k-body interactions: conditional q-normal form of strength functions

For finite quantum many-particle systems modeled with say m fermions in N single particle states and interacting with k-body interactions (k ⩽ m), the wavefunction structure is studied using random matrix theory. The Hamiltonian for the system is chosen to be H = H 0(t) + λV(k) with the unperturbed H 0(t) Hamiltonian being a t-body operator and V(k) a k-body operator with interaction strength λ. Representing H 0(t) and V(k) by independent Gaussian orthogonal ensembles of random matrices in t and k fermion spaces, respectively, the first four moments, in m-fermion spaces, of the strength functions F κ (E) are derived; the strength functions contain all of the information about wavefunction structure. With E denoting the H energies or eigenvalues and κ denoting unperturbed basis states with energy E κ , the F κ (E) give the spreading of the κ states over the eigenstates E. It is shown that the first four moments of F κ (E) are essentially the same as that of the conditional q-normal distribution given in (Szabowski 2010 Electron. J. Probab. 15 1296). This naturally gives asymmetry in F κ (E) with respect to E as E κ increases and also the peak value changes with E κ . Thus, the wavefunction structure in quantum many-fermion systems with k-body interactions in general follows the conditional q-normal distribution.

On the number of equilibria balancing Newtonian point masses with a central force

We consider the critical points (equilibria) of a planar potential generated by n Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular-restricted n-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by n + 1, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in n. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as “ring configurations,” consisting of n − 1 equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as 5n − 5 equilibria. We verify analytically that the ring configuration has at least 5n − 5 equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell’s problem from electrostatics and the image counting problem from gravitational lensing.

Comparison of the magneto-optical properties of the semi-parabolic well with those of the parabolic and rectangular wells under the combined influences of aluminum concentration and hydrostatic pressure

In this paper, we report the comparison of the magneto-optical transport properties of the semi-parabolic potential quantum well with those of the parabolic and rectangular potential quantum wells under the combined influences of the aluminum concentration, the temperature, the hydrostatic pressure, and the Landau levels by using the methods of the quantum field theory for many-particle systems in statistical physics and the profile to calculate analytically the conductivity tensor expression, the magneto-optical absorption power expression of the optically detected magneto-phonon resonance (ODMPR) oscillation, and the full width at half maximum (FWHM) of the ODMPR peak, respectively, in quantum wells with three above different confining potential shapes for typical GaAs/AlxGa1−xAs material. The results obtained from the present study show that (i) the ODMPR peak position undergoes a red-shift as the aluminum concentration and the hydrostatic pressure increase while it experiences a blue-shift as the temperature augments for all three above confining potential shapes; (ii) the FWHM as functions of the barrier's Al-concentration, the system's temperature, and the hydrostatic pressure for all three confining potential shapes is presented to compare; (iii) the dependence of the FWHM on the barrier's Al-concentration and the hydrostatic pressure for the Landau levels for three confining potential shapes also is taken into account in detail; (iv) our present theoretical calculations are found to be consistent with previous experimental calculations.

Low-energy effective quantum field theoretic description of excitations about soliton configurations

Solitons are the classical field configurations connecting two trivial vacua. These are also the solutions of classical field equations of motion with particle-like properties. Moreover, they are localized in space, having finite energy, and are stable against decay into radiation. The coherent state description of kink-solitons is discussed in the present article. Further, the relation between topological solitons and occupation numbers corresponding to low momentum excitations are also discussed coherently. The description of the low energy excitations about solitons in quantum field theory is the main theme of this article. Further, a few physical observables, namely some low order correlation functions, are computed up to certain integral forms. Furthermore, we have shown that it is possible to detect the presence of soliton-like classical configurations in many-particle systems from the nature of the one-point function and non-conservation of momentum feature of one-point, two-point and three-point functions in this low-energy effective field theory of these excitations.

The physics and metaphysics of Tychistic Bohmian Mechanics

The paper takes up Bell's (1987) “Everett (?) theory” and develops it further. The resulting theory is about the system of all particles in the universe, each located in ordinary, 3-dimensional space. This many-particle system as a whole performs random jumps through 3N-dimensional configuration space – hence “Tychistic Bohmian Mechanics” (TBM). The distribution of its spontaneous localisations in configuration space is given by the Born Rule probability measure for the universal wavefunction. Contra Bell, the theory is argued to satisfy the minimal desiderata for a Bohmian theory within the Primitive Ontology framework (for which we offer a metaphysically more perspicuous formulation than is customary). TBM's formalism is that of ordinary Bohmian Mechanics (BM), without the postulate of continuous particle trajectories and their deterministic dynamics. This “rump formalism” receives, however, a different interpretation. We defend TBM as an empirically adequate and coherent quantum theory. Objections voiced by Bell and Maudlin are rebutted. The “for all practical purposes”-classical, Everettian worlds (i.e. quasi-classical histories) exist sequentially in TBM (rather than simultaneously, as in the Everett interpretation). In a temporally coarse-grained sense, they quasi-persist. By contrast, the individual particles themselves cease to persist.

Dicke Transition in Open Many-Body Systems Determined by Fluctuation Effects.

We develop an approach to describe the Dicke transition of interacting many-particle systems strongly coupled to the light of a lossy cavity. A mean-field approach is combined with a perturbative treatment of fluctuations beyond mean field, which becomes exact in the thermodynamic limit. These fluctuations completely change the nature of the steady state, determine the thermal character of the transition, and lead to universal properties of the emerging self-organized states. We validate our results by comparing them with time-dependent matrix-product-state calculations.

Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions

A wave function exposed to measurements undergoes pure state dynamics, with deterministic unitary and probabilistic measurement induced state updates, defining a quantum trajectory. For many-particle systems, the competition of these different elements of dynamics can give rise to a scenario similar to quantum phase transitions. To access it despite the randomness of single quantum trajectories, we construct an $n$-replica Keldysh field theory for the ensemble average of the $n$-th moment of the trajectory projector. A key finding is that this field theory decouples into one set of degrees of freedom that heats up indefinitely, while $n-1$ others can be cast into the form of pure state evolutions generated by an effective non-Hermitian Hamiltonian. This decoupling is exact for free theories, and useful for interacting ones. In particular, we study locally measured Dirac fermions in $(1+1)$ dimensions, which can be bosonized to a monitored interacting Luttinger liquid at long wavelengths. For this model, the non-Hermitian Hamiltonian corresponds to a quantum Sine-Gordon model with complex coefficients. A renormalization group analysis reveals a gapless critical phase with logarithmic entanglement entropy growth, and a gapped area law phase, separated by a Berezinskii-Kosterlitz-Thouless transition. The physical picture emerging here is a pinning of the trajectory wave function into eigenstates of the measurement operators upon increasing the monitoring rate.

Trajectory optimization and maintenance for ascending from the surface of Phobos

• Fuel-optimal ascending trajectories from the surface of Phobos are investigated. • An ERTBP model with the irregular gravity and physical libration is established. • A trajectory maintenance strategy based on the target point method is proposed. • Some trajectories that consume less fuel and are easy to maintain are chosen. The ascending trajectory from the surface of Phobos to a resonant quasi-satellite orbit around Phobos is investigated with a newly proposed dynamical model based on the Elliptic Restricted Three-Body Problem. The proposed model incorporates the non-spherical gravity field and physical libration of Phobos as well. The trajectories from the surface of Phobos are classified into three types according to their z -componentsas short-, middle-, and long-term ascending trajectories. The total Δ V of the two-impulse ascending trajectories is optimized with the particle swarm optimization method. The total Δ V and time of flight of the optimized ascending trajectories are analyzed, and the Pareto Front is refined from the optimized solutions. A multi-impulse maintenance strategy based on the target point method is constructed to ensure that the ascender can insert into the target orbit accurately along the nominal trajectory in the real environment. The robustness of the maintenance strategy is validated by Monte-Carlo simulations in a high-fidelity model, i.e., the N-body problem with the ephemeris, Phobos’ physical libration, non-spherical gravity field of Phobos, and a Gaussian uncertain perturbation. The number of the correction impulses needed is highly positively correlated with the time of flight of the trajectory. Based on the results, middle-term ascending trajectories with less time of flight on the Pareto Front are recommended.

A Comparison of the Georgescu and Vasy Spaces Associated to the N-Body Problems and Applications

We provide new insight into the analysis of N-body problems by studying a compactification $$M_N$$ of $${\mathbb {R}}^{3N}$$ that is compatible with the analytic properties of the N-body Hamiltonian $$H_N$$ . We show that our compactification coincides with a compactification introduced by Vasy using blow-ups in order to study the scattering theory of N-body Hamiltonians and with a compactification introduced by Georgescu using $$C^*$$ -algebras. In particular, the compactifications introduced by Georgescu and by Vasy coincide (up to a homeomorphism that is the identity on $${\mathbb {R}}^{3N}$$ ). Our result has applications to the spectral theory of N-body problems and to some related approximation properties. For instance, results about the essential spectrum, the resolvents, and the scattering matrices of $$H_N$$ (when they exist) may be related to the behavior near $$M_N{\setminus } {\mathbb {R}}^{3N}$$ (i.e., “at infinity”) of their distribution kernels, which can be efficiently studied using our methods. The compactification $$M_N$$ is compatible with the action of the permutation group $$S_N$$ , which allows to implement bosonic and fermionic (anti-)symmetry relations. We also indicate how our results lead to a regularity result for the eigenfunctions of $$H_N$$ .

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

Population cross-diffusion systems of Shigesada–Kawasaki–Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

Slow dynamics of the Fredkin spin chain

The dynamical behavior of a quantum many-particle system is characterized by the lifetime of its excitations. When the system is perturbed, observables of any non-conserved quantity decay exponentially, but those of a conserved quantity relax to equilibrium with a power law. Such processes are associated with a dynamical exponent $z$ that relates the spread of correlations in space and time. We present numerical results for the Fredkin model, a quantum spin chain with a three-body interaction term, which exhibits an unusually large dynamical exponent. We discuss our efforts to produce a reliable estimate $z$=3.16(1) through direct simulation of the quantum evolution and to explain the slow dynamics in terms of an excited bond that executes a constrained random walk in Monte Carlo time.

Total Collisions in the N-Body Shape Space

We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.

Cryptographic Puzzles and Complex Systems

A puzzle lies behind password authentication (PA) and blockchain proof of work (PoW). A cryptographic hash function is commonly used to implement them. The potential problem with secure hash functions is their complexity and rigidity. We explore the use of complex systems constructs such as a cellular automaton (CA) to provide puzzle functionality. The analysis shows that computational irreducibility and sensitivity to initial state phenomena are enough to create simple puzzle systems that can be used for PA and PoW. Moreover, we present puzzle schemata using CA and n-body problems.

Classical and Quantum Mechanical Models of Many-Particle Systems

The collective behaviour of many-particle systems is a common denominator in the challenges of a highly diverse range of applications: from classical problems in Physics (gas dynamics e.g. Boltzmann’s equation, plasma dynamics e.g. various Vlasov equations, semiconductors, quantum mechanics) to current models in biology (kinetic models for collective interaction e.g. swarming, evolution of trait-structured species) to rising topics in social sciences (opinion formation, crowding phenomena) and economics (wealth distribution, mean-field games). Key mathematical questions concern the analysis (global-in-time wellposedness, regularity), rigorous scaling resp. macroscospic limits (model reduction from many-particle models to mean-field/mesoscopic descriptions to macroscopic evolutions), efficient and asymptotic preserving numerical methods and qualitative results (e.g. large-time equilibration).

Localization, Disorder, and Entropy in a Coarse-Grained Model of the Amorphous Solid.

We study the role of disorder in producing the metastable states in which the extent of mass localization is intermediate between that of a liquid and a crystal with long-range order. We estimate the corresponding entropy with the coarse-grained description of a many-particle system used in the classical density functional model. We demonstrate that intermediate localization of the particles results in a change of the entropy from what is obtained from a microscopic approach using for sharply localized vibrational modes following a Debye distribution. An additional contribution is included in the density of vibrational states to account for this excess entropy. A corresponding peak in / vs. frequency matches the characteristic boson peak seen in amorphous solids. In the present work, we also compare the shear modulus for the inhomogeneous solid having localized density profiles with the corresponding elastic response for the uniform liquid in the limit of high frequencies.

Bifurcations of quasi-periodic solutions from relative equilibria in the Lennard\u2013Jones 2-body problem

We propose the general method of proving the bifurcation of new solutions from relative equilibria in N-body problems. The method is based on a symmetric version of Lyapunov center theorem. It is applied to study the Lennard–Jones 2-body problem, where we have proved the existence of new periodic or quasi-periodic solutions.