Narrow Results

The Levy–Levy–Solomon (LLS) model [M. Levy, H. Levy and S. Solomon, Econ. Lett.45, 103 (1994)] is one of the most influential agent-based economic market models. In several publications this model has been discussed and analyzed. Especially Lux and Zschischang [E. Zschischang and T. Lux, Physica A: Stat. Mech. Appl.291, 563 (2001)] have shown that the model exhibits finite-size effects. In this study, we extend existing work in several directions. First, we show simulations which reveal finite-size effects of the model. Second, we shed light on the origin of these finite-size effects. Furthermore, we demonstrate the sensitivity of the LLS model with respect to random numbers. Especially, we can conclude that a low-quality pseudo-random number generator has a huge impact on the simulation results. Finally, we study the impact of the stopping criteria in the market clearance mechanism of the LLS model.

We present a perturbative calculation of finite-size effects near T_c of the φ⁴ lattice model in a d-dimensional cubic geometry of size L with periodic boundary conditions for d>4. The structural differences between the φ⁴ lattice theory and the φ⁴ field theory found previously in the spherical limit are shown to exist also for a finite number of components of the order parameter. The two-variable finite-size scaling functions of the field theory are nonuniversal whereas those of the lattice theory are independent of the nonuniversal model parameters. One-loop results for finite-size scaling functions are derived. Their structure disagrees with the single-variable scaling form of the lowest-mode approximation for any finite ξ/L where ξ is the bulk correlation length. At T_c, the large-L behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to T_c of the lattice model, such as T_max(L) of the maximum of the susceptibility χ, are found to scale asymptotically as T_c-T_max(L) ~L^-d/2, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict χ_max~L^d/2 asymptotically. On a quantitative level, the asymptotic amplitudes of this large-L behavior close to T_c have not been observed in previous MC simulations at d=5 because of nonnegligible finite-size terms ~L^(4-d)/2 caused by the inhomogeneous modes. These terms identify the possible origin of a significant discrepancy between the lowest-mode approximation and previous MC data. MC data of larger systems would be desirable for testing the magnitude of the L^(4-d)/2 and L^4-d terms predicted by our theory.

Boundary condition effects are explored for size-dependent models in thermal equilibrium. Scalar and fermionic models are used for $D=1+3$ (films), $D=1+2$ (hollow cylinder) and $D=1+1$ (ring). For all models, a minimal length is found, below which no thermally-induced phase transition occurs. Using quasiperiodic boundary condition controlled by a contour parameter $𝜃$ ($𝜃=0$ is a periodic boundary condition and $𝜃=1$ is an antiperiodic condition), it results that the minimal length depends directly on the value of $𝜃$. It is also argued that this parameter can be associated to an Aharonov–Bohm phase.

We study Winter or $\delta$-shell model at finite volume (length), describing a small resonating cavity weakly-coupled to a large one. For generic values of the coupling, a resonance of the usual model corresponds, in the finite-volume case, to a compression of the spectral lines; for specific values of the coupling, a resonance corresponds instead to a degenerate or a quasi-degenerate doublet. A secular term of the form $g^{3}N$ occurs in the perturbative expansion of the momenta (or of the energies) of the particle at third order in $g$, where $g$ is the coupling among the cavities and $N$ is the ratio of the length of the large cavity over the length of the small one. These secular terms, which tend to spoil the convergence of the perturbative series in the large volume case, $N\gg 1$, are resummed to all orders in $g$ by means of standard multiscale methods. The resulting improved perturbative expansions provide a rather complete analytic description of resonance dynamics at finite volume.

We discuss finite-size effects in one disordered $\lambda {\phi }^4$ model defined in a $d$-dimensional Euclidean space. We consider that the scalar field satisfies periodic boundary conditions in one dimension and it is coupled with a quenched random field. In order to obtain the average value of the free energy of the system, we use the replica method. We first discuss finite-size effects in the one-loop approximation in $d=3$ and $d=4$. We show that in both cases, there is a critical length where the system develop a second-order phase transition, when the system presents long-range correlations with power-law decay. Next, we improve the above result studying the gap equation for the size-dependent squared mass, using the composite field operator method. We obtain again that the system present a second-order phase transition with long-range correlation with power-law decay.

We present a model to study the effects from external magnetic field, chemical potential and finite size on the phase structures of a massive four- and six-fermion interacting systems. These effects are introduced by a method of compactification of coordinates, a generalization of the standard Matsubara prescription. Through the compactification of the z-coordinate and of imaginary time, we describe a heated system with the shape of a film of thickness L, at temperature ${\beta }^{-1}$ undergoing first- or second-order phase transition. We have found a strong dependence of the temperature transition on the coupling constants $\lambda$ and $\eta$. Besides inverse magnetic catalysis and symmetry breaking for both kinds of transition, we have found an inverse symmetry breaking phenomenon with respect to first-order phase transition.

In this paper, we continue the investigations present in [S. Viaggiu, Physica A473 (2017) 412; 488 (2017) 72.] concerning the spectrum of trapped gravitons in a spherical box, and in particular, inside a Schwarzschild black hole (BH). We explore the possibility that, due to finite size effects, the frequency of the radiation made of trapped gravitons can be modified in such a way that a linear equation-of-state $PV=\gamma U$ for the pressure $P$ and the internal energy $U$ arises. Firstly, we study the case with $U\sim R$, where only fluids with $\gamma >-\frac{1}{3}$ are possible. If corrections $\sim 1/R$ are added to $U$, for $\gamma \in [0,\frac{1}{3}]$, we found no limitation on the allowed value for the areal radius of the trapped sphere $R$. Moreover, for $\gamma >\frac{1}{3}$, we have a minimum allowed value for $R$ of the order of the Planck length $L_P$. Conversely, a fluid with $P<0$ can be obtained but with a maximum allowed value for $R$. With the added term looking like $\sim 1/R$ to the BH internal energy $U$, the well-known logarithmic corrections to the BH entropy naturally emerge for any linear equation-of-state. The results of this paper suggest that finite size effects could modify the structure of graviton’s radiation inside, showing a possible mechanism to transform radiation into dark energy.

In this paper, we consider a simple, purely stochastic model characterized by two conserved quantities (mass density a and energy density h) which is known to display a condensation transition when $h>2a^2$: in the localized phase a single site hosts a finite fraction of the whole energy. Its equilibrium properties in the thermodynamic limit are known and in a recent paper [G. Gotti, S. Iubini and P. Politi, Finite-size localization scenarios in condensation transitions, Phys. Rev. E 103 (2021) 052133] we studied the transition for finite systems. Here, we analyze finite-size effects on the energy distribution and on the relaxation dynamics, showing that extremely large systems should be studied in order to observe the asymptotic distribution and even larger systems should be simulated in order to observe the expected relaxation dynamics.