arXiv subjects

D. S. Dean

Publications and source records attributed to D. S. Dean.

At least 37 records · Page 2Linked to original sources

arXiv2006

Dynamical transition for a particle in a squared Gaussian potential

We study the problem of a Brownian particle diffusing in finite dimensions in a potential given by $ψ= ϕ^2/2$ where $ϕ$ is Gaussian random field. Exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power-law fashion at the dynamical transition temperature. Our results are confronted with numerical simulations where the Gaussian field is constructed, in a standard way, as a sum over random Fourier modes. We show that when the number of Fourier modes is finite the low temperature diffusion constant becomes non-zero and has an Arrhenius form. Thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition. In addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model.

cond-mat.dis-nn↗

arXiv2006

Fluctuation induced interactions between domains in membranes

We study a model lipid bilayer composed of a mixture of two incompatible lipid types which have a natural tendency to segregate in the absence of membrane fluctuations. The membrane is mechanically characterized by a local bending rigidity $κ(ϕ)$ which varies with the average local lipid composition $ϕ$. We show, in the case where $κ$ varies weakly with $ϕ$, that the effective interaction between lipids of the same type can either be everywhere attractive or can have a repulsive component at intermediate distances greater than the typical lipid size. When this interaction has a repulsive component, it can prevent macro-phase separation and lead to separation in mesophases with a finite domain size. This effect could be relevant to certain experimental and numerical observations of mesoscopic domains in such systems.

cond-mat.soft↗

arXiv2005

Some observations on the renormalization of membrane rigidity by long-range interactions

We consider the renormalization of the bending and Gaussian rigidity of model membranes induced by long-range interactions between the components making up the membrane. In particular we analyze the effect of a finite membrane thickness on the renormalization of the bending and Gaussian rigidity by long-range interactions. Particular attention is paid to the case where the interactions are of a van der Waals type.

cond-mat.soft↗

arXiv2005

The field theory of symmetrical layered electrolytic systems and the thermal Casimir effect

We present a general extension of a field-theoretic approach developed in earlier papers to the calculation of the free energy of symmetrically layered electrolytic systems which is based on the Sine-Gordon field theory for the Coulomb gas. The method is to construct the partition function in terms of the Feynman evolution kernel in the Euclidean time variable associated with the coordinate normal to the surfaces defining the layered structure. The theory is applicable to cylindrical systems and its development is motivated by the possibility that a static van der Waals or thermal Casimir force could provide an attractive force stabilising a dielectric tube formed from a lipid bilayer, an example of which are t-tubules occurring in certain muscle cells. In this context, we apply the theory to the calculation of the thermal Casimir effect for a dielectric tube of radius $R$ and thickness $δ$ formed from such a membrane in water. In a grand canonical approach we find that the leading contribution to the Casimir energy behaves like $-k_BTLκ_C/R$ which gives rise to an attractive force which tends to contract the tube radius. We find that $κ_C \sim 0.3$ for the case of typical lipid membrane t-tubules. We conclude that except in the case of a very soft membrane this force is insufficient to stabilise such tubes against the bending stress which tend to increase the radius. We briefly discuss the role of lipid membrane reservoir implicit in the approach and whether its nature in biological systems may possibly lead to a stabilising mechanism for such lipid tubes.

cond-mat.soft↗

arXiv2004

The thermal Casimir effect in lipid bilayer tubules

We calculate the thermal Casimir effect for a dielectric tube of radius $R$ and thickness delta formed from a membrane in water. The method uses a field-theoretic approach in the grand canonical ensemble. The leading contribution to the Casimir free energy behaves as -k_BTL kappa_C/R giving rise to an attractive force which tends to contract the tube. We find that kappa_C ~ 0.3 for the case of typical lipid membrane t-tubules. We conclude that except in the case of a very soft membrane this force is insufficient to stabilize such tubes against the bending stress which tends to increase the radius.

cond-mat.soft↗

arXiv2004

Perturbation theory for the effective diffusion constant in a medium of random scatterer

We develop perturbation theory and physically motivated resummations of the perturbation theory for the problem of a tracer particle diffusing in a random media. The random media contains point scatterers of density $ρ$ uniformly distributed through out the material. The tracer is a Langevin particle subjected to the quenched random force generated by the scatterers. Via our perturbative analysis we determine when the random potential can be approximated by a Gaussian random potential. We also develop a self-similar renormalisation group approach based on thinning out the scatterers, this scheme is similar to that used with success for diffusion in Gaussian random potentials and agrees with known exact results. To assess the accuracy of this approximation scheme its predictions are confronted with results obtained by numerical simulation.

cond-mat.dis-nn↗

arXiv2004

Self diffusion in a system of interacting Langevin particles

The behavior of the self diffusion constant of Langevin particles interacting via a pairwise interaction is considered. The diffusion constant is calculated approximately within a perturbation theory in the potential strength about the bare diffusion constant. It is shown how this expansion leads to a systematic double expansion in the inverse temperature $β$ and the particle density $ρ$. The one-loop diagrams in this expansion can be summed exactly and we show that this result is exact in the limit of small $β$ and $ρβ$ constant. The one-loop result can also be re-summed using a semi-phenomenological renormalization group method which has proved useful in the study of diffusion in random media. In certain cases the renormalization group calculation predicts the existence of a diverging relaxation time signalled by the vanishing of the diffusion constant -- possible forms of divergence coming from this approximation are discussed. Finally, at a more quantitative level, the results are compared with numerical simulations, in two-dimensions, of particles interacting via a soft potential recently used to model the interaction between coiled polymers.

cond-mat.soft↗

arXiv2003

Field theoretic calculation of the surface tension for a model electrolyte system

We carry out the calculation of the surface tension for a model electrolyte to first order in a cumulant expansion about a free field theory equivalent to the Debye-Hückel approximation. In contrast with previous calculations, the surface tension is calculated directly without recourse to integrating thermodynamic relations. The system considered is a monovalent electrolyte with a region at the interface, of width h, from which the ionic species are excluded. In the case where the external dielectric constant epsilon_0 is smaller than the electrolyte solution's dielectric constant epsilon we show that the calculation at this order can be fully regularized. In the case where h is taken to be zero the Onsager-Samaras limiting law for the excess surface tension of dilute electrolyte solutions is recovered, with corrections coming from a non-zero value of epsilon_0/epsilon.

cond-mat.soft↗

arXiv2003

Effective diffusion constant in a two dimensional medium of charged point scatterers

We obtain exact results for the effective diffusion constant of a two dimensional Langevin tracer particle in the force field generated by charged point scatterers with quenched positions. We show that if the point scatterers have a screened Coulomb (Yukawa) potential and are uniformly and independently distributed then the effective diffusion constant obeys the Volgel-Fulcher-Tammann law where it vanishes. Exact results are also obtained for pure Coulomb scatterers frozen in an equilibrium configuration of the same temperature as that of the tracer.

cond-mat.dis-nn↗

arXiv2003

The field theoretic derivation of the contact value theorem in planar geometries and its modification by the Casimir effect

The contact value theorem for Coulomb gases in planar or film-like geometries is derived using a Hamiltonian field theoretic representation of the system. The case where the film is enclosed by a material of different dielectric constant to that of the film is shown to contain an additional Casimir-like term which is generated by fluctuations of the electric potential about its mean-field value.

cond-mat.soft↗

arXiv2002

A possible experimental test of the thermodynamic approach to granular media

We study the steady state distribution of the energy of the Sherrington-Kirkpatrick model driven by a tapping mechanism which mimics the mechanically driven dynamics of granular media. The dynamics consists of two phases: a zero temperature relaxation phase which leads the system to a metastable state, then a tapping which excites the system thus reactivating the relaxational dynamics. Numerically we investigate whether the distribution of the energies of the blocked states obtained agrees with a simple canonical form of the Edwards measure. It is found that this canonical measure is in good agreement with the dynamically measured energy distribution. A possible experimental test of the Edwards measure based on the study here is proposed.

cond-mat.dis-nn↗

arXiv2002

The number of metastable states in the generalized random orthogonal model

We calculate the number of metastable states in the generalized random orthogonal model. The results obtained are verified by exact numerical enumeration for small systems sizes but taking into account finite size effects. These results are compared with those for Hopfield model in order to examine the effect of strict orthonormality of neural network patterns on the number of metastable states.

cond-mat.dis-nn↗

arXiv2002

The role of the interaction matrix in mean-field spin glasses

Mean-field models of 2-spin Ising spin glasses with interaction matrices taken from ensembles which are invariant under O(N) transformations are studied. A general study shows that the nature of the spin glass transition can be deduced from the eigenvalue spectrum of the interaction matrix. A simple replica approach is derived to carry out the average over the O(N) disorder. The analytic results are confirmed by extensive Monte Carlo simulations for large system sizes and by exact enumeration for small system sizes.

cond-mat.dis-nn↗

arXiv2002

Weak non-linear surface charging effects in electrolytic films

A simple model of soap films with nonionic surfactants stabilized by added electrolyte is studied. The model exhibits charge regularization due to the incorporation of a physical mechanism responsible for the formation of a surface charge. We use a Gaussian field theory in the film but the full non-linear surface terms which are then treated at a one-loop level by calculating the mean-field Poisson-Boltzmann solution and then the fluctuations about this solution. We carefully analyze the renormalization of the theory and apply it to a triple layer model for a thin film with Stern layer of thickness $h$. For this model we give expressions for the surface charge $σ(L)$ and the disjoining pressure $P_d(L)$ and show their dependence on the parameters. The influence of image charges naturally arise in the formalism and we show that predictions depend strongly on $h$ because of their effects. In particular, we show that the surface charge vanishes as the film thickness $L \to 0$. The fluctuation terms about this class of theories exhibit a Casimir-like attraction across the film and although this attraction is well known to be negligible compared with the mean-field component for thick films in the presence of electrolyte, in the model studied here these fluctuations also affect the surface charge regulation leading to a fluctuation component in the disjoining pressure which has the same behavior as the mean-field component even for large film thickness.

cond-mat.soft↗

arXiv2002

Electrostatic Fluctuations in Soap Films

A field theory to describe electrostatic interactions in soap films, described by electric multi-layers with a generalized thermodynamic surface-charging mechanism, is studied. In the limit where the electrostatic interactions are weak this theory is exactly soluble. The theory incorporates in a consistent way, the surface-charging mechanism and the fluctuations in the electrostatic field which correspond to the zero frequency component of the van der Waals force. It is shown that these terms lead to a Casimir like attraction which can be sufficiently large to explain the transition between the Common Black Film to a Newton Black Film.

cond-mat.stat-mech↗

arXiv2001

The Nearest and Next Nearest Neighbor Sherrington-Kirkpatrick Model

The mean field theory of a spin glass with a specific form of nearest and next nearest neighbor interactions is investigated. Depending on the sign of the interaction matrix chosen we either find the continuous replica symmetry breaking seen in the Sherrington-Kirkpartick model or a one step solution similar to that found in structural glasses. Our results are confirmed by numerical simulations and the link between the type of spin-glass behavior and the density of eigenvalues of the interaction matrix is discussed.

cond-mat.dis-nn↗

arXiv2001

Phase transitions in the steady state behavior of mechanically perturbed spin glasses and ferromagnets

We analyze the steady state regime of systems interpolating between spin glasses and ferromagnets under a tapping dynamics recently introduced by analogy with the dynamics of mechanically perturbed granular media. A crossover from a second order to first order ferromagnetic transition as a function of the spin coupling distribution is found. The flat measure over blocked states introduced by Edwards for granular media is used to explain this scenario. Annealed calculations of the Edwards entropy are shown to qualitatively explain the nature of the phase transitions. A Monte-Carlo construction of the Edwards measure confirms that this explanation is also quantitatively accurate.

cond-mat.stat-mech↗

arXiv2001

Extreme Value Statistics of Hierarchically Correlated Variables: Deviation from Gumbel Statistics and Anomalous Persistence

We study analytically the distribution of the minimum of a set of hierarchically correlated random variables $E_1$, $E_2$, $...$, $E_N$ where $E_i$ represents the energy of the $i$-th path of a directed polymer on a Cayley tree. If the variables were uncorrelated, the minimum energy would have an asymptotic Gumbel distribution. We show that due to the hierarchical correlations, the forward tail of the distribution of the minimum energy becomes highly nnon universal, depends explicitly on the distribution of the bond energies $ε$ and is generically different from the super-exponential forward tail of the Gumbel distribution. The consequence of these results to the persistence of hierarchically correlated random variables is discussed and the persistence is also shown to be generically anomalous.

cond-mat.stat-mech↗