Φ4 Lattice Research Articles

Validity of local thermal equilibrium in anomalous heat diffusion

Local thermal equilibrium (LTE) is a general presumption of many theoretical analyses in nonequilibrium statistical physics. It describes a situation that although the system is not in global thermal equilibrium, each small portion of the system may still be described approximately by the laws of thermal equilibrium. The validity of LTE has however seldom been investigated carefully. Here, by studying the ensemble velocity distribution and its spatial correlation, we present strong evidences for the lack of LTE in anomalous heat diffusion processes in one dimensional harmonic lattices and the Fermi–Pasta–Ulam-β lattices. In particular, clear nonzero excess kurtosis and long range correlations have been observed, with values scaling linearly with the initial temperature difference. Therefore near thermal equilibrium is not sufficient to achieve LTE. Some existing studies that are based on the existence of LTE should be revisited. Using the same methods, we also show that LTE is still valid in the ϕ4 lattice, in which heat diffuses following Fourier’s law.

Negative differential thermal resistance phenomenon in the FK-ϕ4 lattices

Negative differential thermal resistance (NDTR) was investigated in a system consisting of two dissimilar anharmonic lattices exemplified by Frenkel–Kontorova (FK) lattices and ϕ4 lattices (FK-ϕ4). The results indicate that: (i) For appropriate periodic on-site potentials of FK lattice, as the system is asymmetric, an NDTR phenomenon will appear. (ii) The NDTR phenomenon also depends on the quartic on-site potentials of ϕ4 lattice as the other parameters remain unchanged. With the increment of the quartic on-site potential, the NDTR phenomenon will gradually disappear. (iii) As the system is asymmetric, the averaged environmental reference temperature and coupling displacement cannot only enhance NDTR phenomenon but also regulate thermal rectifier of the system as a thermal switch. (IV) Along with an increasing atomic number of the system, the NDTR phenomenon gradually disappears. Our results indicate that the on-site potential of nonlinear lattices plays a crucial role in the design of thermal devices.

Transitions to Equilibrium State in Classical ϕ4 Lattice**The project supported by National Natural Science Foundation of China, the Nonlinear Science Project of China, and the Foundation of Doctoral Training of the Educational Ministry of China

Statistical behavior of a classical ϕ4 Hamiltonian lattice is investigated from microscopic dynamics. The largest Lyapunov exponent and entropies are considered for manifesting chaos and equipartition behaviors of the system. It is found, for the first time, that for any large while finite system size there exist two critical couplings for the transitions to equipartitions, and the scaling behaviors of these lower and upper critical couplings vs. the system size are numerically obtained.

Lattice φ4 Theory of Finite-Size Effects Above the Upper Critical Dimension

We present a perturbative calculation of finite-size effects near Tc 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 Tc, the large-L behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to Tc of the lattice model, such as T max (L) of the maximum of the susceptibility χ, are found to scale asymptotically as Tc-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 ~Ld/2 asymptotically. On a quantitative level, the asymptotic amplitudes of this large-L behavior close to Tc 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 L4-d terms predicted by our theory.

Nontrivial fixed point in the 4D Φ4 lattice model with internal O(N) symmetry

It is shown that the infinite dimensional critical surface of general euclidean lattice actions in a generic four-dimensional scalar field theory with $\Phi^4$ interactions has a domain of special multicritical points where higher dimensional operators play a special role. Renormalized trajectories of higher derivative continuum field theories with nontrivial interactions are traced back to special ultraviolet stable fixed points on the manifold of multicritical points. These fixed points have an infinite number of relevant directions which identify the universality classes of critical higher derivative field theories. The relevance of the new fixed point structure is discussed within the context of the triviality Higgs mass bound.

The random-walk representation of classical spin systems and correlation inequalities

We use the random-walk representation to prove the first few of a new family of correlation inequalities for ferromagnetic ϕ4 lattice models. These inequalities state that the finite partial sums of the propagator-resummed perturbation expansion for the 4-point function form an alternating set of rigorous upper and lower bounds for the exact 4-point function. Generalizations to 2n-point functions are also given. A simple construction of the continuum ϕ quantum field theory (d<4), based on these inequalities, is described in a companion paper.

Critical exponents and elementary particles

Particles are shown to exist for a.e. value of the mass in single phase φ4 lattice and continuum field theories and nearest neighbor Ising models. The particles occur in the form of poles at imaginary (Minkowski) momenta of the Fourier transformed two point function. The new inequalitydm2/dσ≦Z, where σ=m02 is a bare mass2 andZ is the strength of the particle pole, is basic to our method. This inequality implies inequalities for critical exponents.

The Ising model limit of 𝜙⁴ lattice fields

We show that the λ → ∞ \lambda \to \infty limit of λ ϕ 4 \lambda {\phi ^4} lattice fields is an Ising model.