Scaling theory for nonequilibrium systems

We develop a general scaling theory of one-dimensional systems withN components having applications to disorder-order-transitions or order-order transitions of non-equilibrium systems, such as lasers, hydrodynamical systems and non-equilibrium chemical reactions. We include both cases of soft and hard modes. Since fluctuations play a decisive role at the transition point, we take fully account of them. We start from general equations of motion which contain nonlinear forces (or rates), diffusion terms and fluctuating forces. These equations depend on external parameters. When linearized around their steady state solutions, the equations allow for stable, marginal or unstable solutions. The solutions near critical points are represented as superpositions of marginal solutions, whose amplitudes are determined by comparing the coefficients of the scaling parameter up to third order. The scaling of the fluctuating forces and, in the case of chemical reactions, their correlation functions are derived in detail.     

Scaling theory of quantum resistance distributions in disordered systems

We have derived explicitly, the large scale distribution of quantum Ohmic resistance of a disordered one-dimensional conductor. We show that in the thermodynamic limit this distribution is characterized by two independent parameters for strong disorder, leading to a two-parameter scaling theory of localization. Only in the limit of weak disorder we recover single parameter scaling, consistent with existing theoretical treatments.     

An evolution theory in finite size systems

A new model of evolution is presented for finite size systems. Conditions under which a minority species can emerge, spread and stabilize to a macroscopic size are studied. It is found that space organization is instrumental in addition to a qualitative advantage. Some peculiar topologies ensure the overcome of the initial majority species. However the probability of such local clusters is very small and depend strongly on the system size. A probabilistic phase diagram is obtained for small sizes. It reduces to a trivial situation in the thermodynamic limit, thus indicating the importance of dealing with finite systems in evolution problems. Results are discussed with respect to both Darwin and punctuated equilibria theories. Received 25 June 2000     

Conductance of finite systems and scaling in localization theory

The conductance of finite systems plays a central role in the scaling theory of localization (Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)). Usually it is defined by the Landauer-type formulas, which remain open the following questions: (a) exclusion of the contact resistance in the many-channel case; (b) correspondence of the Landauer conductance with internal properties of the system; (c) relation with the diffusion coefficient D(??, q) of an infinite system. The answers to these questions are obtained below in the framework of two approaches: (1) self-consistent theory of localization by Vollhardt and Wölfle, and (2) quantum mechanical analysis based on the shell model. Both approaches lead to the same definition for the conductance of a finite system, closely related to the Thouless definition. In the framework of the self-consistent theory, the relations of finite-size scaling are derived and the Gell-Mann-Low functions ??(g) for space dimensions d = 1, 2, 3 are calculated. In contrast to the previous attempt by Vollhardt and Wölfle (1982), the metallic and localized phase are considered from the same standpoint, and the conductance of a finite system has no singularity at the critical point. In the 2D case, the expansion of ??(g) in 1/g coincides with results of the ??-model approach on the two-loop level and depends on the renormalization scheme in higher loops; the use of dimensional regularization for transition to dimension d = 2 + ?? looks incompatible with the physical essence of the problem. The results are compared with numerical and physical experiments. A situation in higher dimensions and the conditions for observation of the localization law ??(??) ?? ?i?? for conductivity are discussed.     

Meson exchange and the theory of finite fermi systems

The spin-dependent parameters g₀ and g'₀ in Migdal's theory of finite Fermi systems are studied. Assuming their values to be such as to reproduce the M1 states in ²⁰⁸Pb at 7 and 8 MeV, the origin of these (large) values in meson-exchange contributions to the nucleon-nucleon force is studied. Pion exchange contributes little, because of cancellation between matrix elements from the spin-spin and tensor components. The largest, nearly equal, contributions to g₀ and g'₀ come from p-meson exchange. In order to explain the large values needed for g₀ and g'₀, the tensor coupling of the ρ-meson must be substantially larger than the effective ones found in current phenomenological potentials, such as the Reid soft-core and Hamada-Johnston ones. The large value of the coupling needed follows from work analyzing nucleon form factors.     

Self-consistent theory of finite Fermi systems and radii of nuclei

Present-day self-consistent approaches in nuclear theory were analyzed from the point of view of describing distributions of nuclear densities. The generalized method of the energy density functional due to Fayans and his coauthors (this is the most successful version of the self-consistent theory of finite Fermi systems) was the first among the approaches under comparison. The second was the most successful version of the Skyrme-Hartree-Fock method with the HFB-17 functional due to Goriely and his coauthors. Charge radii of spherical nuclei were analyzed in detail. Several isotopic chains of deformed nuclei were also considered. Charge-density distributions ρ _ch(r) were calculated for several spherical nuclei. They were compared with model-independent data extracted from an analysis of elastic electron scattering on nuclei.     

Some problems in the generalized theory of finite Fermi systems

A realistic version of the generalization of the theory of finite Fermi systems to the case where some complex configurations involving phonons are explicitly taken into account is proposed. Secular equations describing the fragmentation of simple states in odd and even-even nuclei over complex configurations that belong to, respectively, the quasiparticle ? phonon + quasiparticle ? phonon ? phonon and the two quasiparticles ? phonon type and which are presently of greatest interest are derived on the basis of general relations for nuclei that involve pairing (nonmagic nuclei). These equations take into account effects associated with ground-state correlations due to complex configurations and with the additional quasiparticle-phonon mechanism of Cooper pairing in nuclei. The effects in question were disregarded previously, but they are of interest since they can be observed in present-day experiments.     

Scaling approach to the theory of frequency dependent hopping conductivity of disordered systems

By studying the change of the probability distribution function for the hopping propagation of carriers under differential changes of the temperature and density of sites partial differential equations are derived which couple the frequency, temperature, and density dependence of the conductivity. In certain special cases the conductivity can be calculated explicitly for zero frequency as well as in regions where the frequency dependence is known from experiment or some other theory. This leads to results which are well suited for experimental tests. Approximations used in this treatment are discussed and related to other types of theories for hopping conductivity.     

Closed-orbit theory of spatial density oscillations in finite fermion systems

We investigate the particle and kinetic-energy densities for N noninteracting fermions confined in a local potential. Using Gutzwiller's semiclassical Green function, we describe the oscillating parts of the densities in terms of closed nonperiodic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions and a "local virial theorem" valid also for arbitrary nonintegrable potentials. We give simple analytical formulas for the density oscillations in a one-dimensional potential.     

Scaling in the sine-Gordon theory

It is shown that both the classical and the quantum sine-Gordon theory depend on a single scaling parameter and therefore the coupling constant cannot be freely chosen. To introduce a meaningful coupling constant it is proposed to include higher Fourier terms in the sine-Gordon potential. The two term case is exactly solvable.     

Scaling theory of polymer thermodiffusion

The motion of a linear polymer chain in a good solvent under a temperature gradient is examined theoretically by breaking up the flexible chain into Brownian rigid rods, and writing down an equation of motion for each rod. The motion is driven by two forces. The first one is Waldmann’s thermophoretic force (stemming from the departure of the solvent’s molecular-velocity distribution from Maxwell’s equilibrium distribution) which here is extrapolated to a dense medium. The second force is due to the fact that the viscous friction varies with position owing to the temperature gradient, which brings an important correction to the Stokes law of friction. We use scaling considerations relying upon disparate length scales and omitting non-universal numerical prefactors. The present scaling theory is compared with recent experiments on the thermodiffusion of polymers and is shown to account for (i) the existence of both signs of the thermodiffusion coefficient of long chains, (ii) the order of magnitude of the coefficient, (iii) its independence of the chain length in the high-polymer limit and (iv) its dependence on the solvent viscosity.     

The bond problem in a one-dimensional percolation theory for finite systems

The results of investigations of main characteristics of a one-dimensional percolation theory (percolation threshold, critical exponents of correlation radius and specific heat, and free energy) are presented for the bond and site problems. For the first time it is shown that for a finite-size system the stability condition is fulfilled while the scaling hypothesis is inacceptable for one-dimensional bond problem.     

Scaling theory of mixed amphiphilic monolayers

We predict the elastic properties of mixed amphiphilic monolayers in the swollen state within the blob model using scaling arguments. First the elastic moduli and the spontaneous curvature of a bimodal brush are determined as a function of the composition and the relative chain length. We obtain simple and useful scaling functions which interpolate between the elastic moduli of a pure short-chain brush and a pure long-chain brush. By using the analogy between block copolymer interfaces and polymeric brushes, the effect of mixing on self-assembled diblock copolymer monolayers is investigated in the swollen state. We calculate various interfacial properties, such as the equilibrium surface coverage, interface curvature, and the mixing free energy as a function of the composition. In general, we find a nonlinear dependence on the composition, which deviates from the simple linear averaging of the properties of pure components. Our results are used to discuss a recent experiment on the effect of amphiphilic block copolymers on the efficiency of microemulsions. Received 29 December 2000 and Received in final form 19 March 2001     

Scaling theory for unipolar resistance switching

We investigate a reversible percolation system showing unipolar resistance switching in which percolating paths are created and broken alternately by the application of an electric bias. Owing to the dynamical changes in the percolating paths, different from those in classical percolating paths, a detailed understanding of the structure is demanding and challenging. Here, we develop a scaling theory that can explain the transport properties of these conducting paths; the theory is based on the fractal geometry of a percolating cluster. This theory predicts that two scaling behaviors emerge, depending on the topologies of the conducting paths. We confirm these theoretical predictions experimentally by observing material-independent universal scaling behaviors in unipolar resistance switching.