Mean field theory for Coulomb systems

We study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero. We prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field(x) where is given by-+ 2z sinh() =q This is the mean field equation of Debye and Hückel. The proof uses the sine-Gordon transformation, the Mayer expansion, and a correlation inequality.Work partially supported by NSF Grant MCS 82-02115.     

Mean field theory of self-organized critical phenomena

A mean field theory is presented for the recently discovered self-organized critical phenomena. The critical exponents are calculated and found to be the same as the mean field values for percolation. The power spectrum has 1/f behavior with exponentg4=1.     

Mean field theory for nonequilibrium network reconstruction

There has been recent progress on inferring the structure of interactions in complex networks when they are in stationary states satisfying detailed balance, but little has been done for nonequilibrium systems. Here we introduce an approach to this problem, considering, as an example, the question of recovering the interactions in an asymmetrically coupled, synchronously updated Sherrington-Kirkpatrick model. We derive an exact iterative inversion algorithm and develop efficient approximations based on dynamical mean-field and Thouless-Anderson-Palmer equations that express the interactions in terms of equal-time and one-time-step-delayed correlation functions.     

Mean field theory and geodesics in general relativity

It is proven that all geodesics in a mean gravitational field can be interpreted locally as the averages of geodesics in the unaveraged field. The respective time-like, space-like or null character of averaged and unaveraged geodesics is discussed carefully. Finally, some important astrophysical and cosmological applications and consequences are investigated.     

Mean field theory for compositionally modulated magnetic superlattices

We study a two-constituent ferromagnetic superlattice modulated compositionally. It is found that, within the molecular field approximation and in the continuum limit, the Curie temperature varies linearly with wavevector for a sinusoidal modulation.     

Mean field theory of a quantum heisenberg spin glass

A full mean-field solution of a quantum Heisenberg spin-glass model is presented in a large- N limit. A spin-glass transition is found for all values of the spin S. The quantum critical regime associated with the quantum transition at S = 0 and the various regimes in the spin-glass phase at high spin are analyzed. The specific heat is shown to vanish linearly with temperature. In the spin-glass phase, intriguing connections between the equilibrium properties of the quantum problem and the out-of-equilibrium dynamics of classical models are pointed out.     

Mean field theory of dynamic phase transitions in ferromagnets

We have studied the second order dynamic phase transition (DPT) of the two-dimensional kinetic Ising model by means of numerical calculations. While it is well established that the order parameter Q of the DPT is the average magnetization per external field oscillation cycle, the possible identity of the conjugate field has been addressed only recently. In this work, we demonstrate that our entire set of numerical data is fully consistent with the applied bias field H_b being the conjugate field of order parameter Q. For this purpose, we have analyzed the Q(H_b)-dependence and we have found that it follows the expected power law behavior with the same critical exponent as the mean field equilibrium case.     

Mean field analysis of aZ 2-asymmetric gauge theory

Using mean field techniques we study aZ ₂-gauge theory with asymmetric couplings for two kinds of plaquettes. CalculatingO(1/d) corrections to the mean field approximation we obtain a phase diagram that agrees with Monte Carlo data. In addition, various points where equivalent continuum theories arise are analyzed within this approximation.     

Mean field theory and ϵ-expansion for Anderson localization

We present a field theoretic formulation of Anderson localization of an electron in a random potential. In mean field theory we find a mobility edge at energy E_c separating a region with no states from one with conducting states. When the nearest neighbor hopping is a random variable with variance σ, E_c²=4zσ², where z is the coordination number. We study this mobility edge using the ?-expansion. We find an upper critical dimension of eight near which this mobility transition is in the same universality class as the statistics of dilute branched polymers (lattice animals).     

Mean field theory of directed polymers with random complex weights

We show that for the problem of directed polymers on a tree with i.i.d. random complex weights on each bond, three possible phases can exist; the phase of a particular system is determined by the distribution of the random weights. For each of these three phases, we give the expression of the free energy per unit length in the limit of infinitely long polymers. Our proofs require several hypotheses on the distribution , most importantly, that the amplitude and the phase of each complex weight be statistically independent. The main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy. We illustrate our results by some examples and discuss possible generalizations to a larger class of distributions, to Random Energy Models, and to the finite dimensional case. We note that our results are not in agreement with the predictions of a recent replica approach to a similar problem.     

Mean field theory and monte carlo simulation of multisite adsorption

Multisite or $\text{1}\text{n}$ adsorption on single crystal surfaces takes place when the adatom diameter is greater than the distance between two neighbouring adsorption sites. The adsorption isotherm has been deduced using a mean field approximation. The same problem was studied by Monte Carlo simulation. The results show asymmetric adsorption isotherms with respect to a degree of coverage of 0.5. The critical temperature of a first order phase transition is increased as compared with the case of $\text{1}\text{1}$ adsorption. The theoretical and the simulated isotherms are in a good agreement for low adatom-adatom attraction. The theoretical and simulated isotherms obtained were used for interpretation of experimental results on electrolytic underpotential adsorption of Tl and Pb on Ag(100) faces.     

Mean field calculations and defect gases in lattice gauge theory

We investigate the relationship between local defects and the mean field method in lattice gauge theory. In particular we clarify the role of defects in establishing the equivalence between mean field calculations with and without gauge fixing. In two dimensions we derive the area law for the Wilson loop by a mean field calculation incorporating defects. We also establish a general rule about mean field variables which are appropriate to handle defects induced by an action that almost possesses a local symmetry group, and we apply it to theZ(2) Higgs model and to the mixedSU(2)-SO(3) model.     

A quantum field theory of lattice dynamics and melting in compounds

Summary A unified theory on the lattice dynamics and the melting in compounds is presented by taking into account the Coulomb interaction. The melting temperature is determined by the mean value of the eigenfrequencies of the atoms. The theory is applied to zincblende and rock salt structure crystals.

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Riassunto Si presenta una teoria unificata sulla dinamica dei reticoli e sulla fusione nei composti considerando l’interazione di Coulomb. La temperatura di fusione è determinata dal valore medio delle autofrequenze degli atomi. Si applica la teoria alla zinco-blenda e a cristalli con struttura del tipo di sali di rocca.

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Резюме Предлагается теория динамики решетки и плавления в химических соединениях, которя учитывает кулоновское взаимодействие. Температупа плавления определяется, как средияя величина собстенных чатстот атомов. Предложенная теория применяется к кристаллам со структурой цинковой обманки и кчменной соли.

     

A quantum field theory of lattice dynamics and melting in fcc crystals

Phonons in crystals are constructed by particle-hole excitations of ion cores. The Ward-Takahashi relations play an essential role in constructing the lattice dynamics and determining the melting temperature. The phonon dispersion and the melting temperature in fcc crystals are presented. In principle, our theory covers metallic, covalent and molecular crystals.     

A quantum field theory of lattice dynamics and melting in nonprimitive crystals

A unified theory of lattice dynamics and melting is presented from the view-point of quantum field theory at finite temperature. The lattice dynamics and the melting in nonprimitive crystals which contain more than one atom in a unit cell are investigated in the random phase approximation using the two-bands model for atoms. The gap equation for atoms plays an essential role in constructing the lattice dynamics and determining the melting temperature. Our theory is applied to sphalerite and diamond structure in the nearest-neighbour approximation for the interatomic potentials. Our theory covers all types of crystals.     

Surface theory of melting

We demonstrate that melting is a surface initiated process. The surface becomes unstable before the bulk and the process of melting consists in the unstable surface that proceeds into the otherwise stable bulk.     

Pion condensation at finite temperature: (I). Mean field theory

The pion-condensed state of neutron-rich matter at finite temperature is calculated within the framework of a simple σ-model, treating the pion field as a mean field. At high densities the matter is condensed with a spatially non-uniform condensate. However, we find the unexpected result that as the density is lowered, at any finite temperature, pure neutron matter always makes a transition to a state with a spatially uniform condensate. Pure neutron matter, within mean field theory, is condensed at all non-zero temperatures and densities. Matter with a small proton fraction at zero temperature has a qualitatively similar phase diagram, except that it is normal when both the temperature and density are sufficiently low.