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Papers by John Kozak

Research paper thumbnail of Maine South Basketball Varsity Roster 2012-2013 # Name Ht. Wt. Year Position

Research paper thumbnail of Comparative Study of Spontaneous Emission and Spin-Lattice Relaxation atT=0

Physical Review B, 1971

In this paper we present a comparative study of spontaneous emission and spin-lattice relaxation ... more In this paper we present a comparative study of spontaneous emission and spin-lattice relaxation at zero temperature. In particular, we study the time evolution of the density matrix for two simple models as determined from an analysis of the Prigogine-Rdsibois master equation. The first model treated is that of the Wigner-Weisskopf atom in a three-dimensional radiation field; the second model is that of a single, effective spin in interaction with the phonon modes of a three-dimensional lattice. The divergence which arises in the solution of the master equation for the first model is avoided using a frequency cutoff. A frequency cutoff in the second model is imposed by the upper bound of the spectrum of modes in the crystal, and this fact manifests itself when one integrates over the first Brillouin zone only. From a detailed numerical study of the analytic results obtained in solving the master equation, we find that for both models the relaxation to equilibrium is characterized, in part, by a sequence of slowly damped oscillations. This result seems to be in agreement with the observation made by Zwanzig, namely, that exponential decay in time seems not to be universal, and may, in fact, be hidden behind some other kind of time dependence. The numerical study also reveals, however, that the nonexponential modes of decay can be quantitatively different in magnitude and qualitatively different in structure for atomic versus spin systems. Finally, based on the solution obtained for the spin problem, an estimate is made of the relaxation time for cerium ethyl sulfate, and this estimate is found to be consistent with experiment.

Research paper thumbnail of Method for continuously casting a strip of alloyed metal

Research paper thumbnail of Critical Behavior of the Yvon-Born-Green Equation: Effects of Dimensionality

Physical Review Letters, 1982

Analytic studies of the Yvon-Born-Green equation have suggested that it may show no true critical... more Analytic studies of the Yvon-Born-Green equation have suggested that it may show no true critical region for spatial dimension d &4 but that for d & 4 there is a critical point of the mean-field type. Presented here are the results of numerical solution of the equation which strongly support these suggestions. Thus the Yvon-Born-Green equation exhibits the correct borderline dimensionality of four and the expected mean-field behavior for d 4, but for d & 4 has only a region of long, but finite, ranged correlations.

Research paper thumbnail of Neuropsychological Performance in Psychiatric Patients with or Without Documented Brain Dysfunction

International Journal of Neuroscience, 1988

The current study examined neuropsychological performance of schizophrenic and depressed patients... more The current study examined neuropsychological performance of schizophrenic and depressed patients with and without structural or EEG signs of brain dysfunctioning. The neuropsychological test battery was designed to sample intellectual functioning, psychomotor skills, nonverbal memory, and novel problem solving ability. Patients were classified into four groups: schizophrenics without signs of brain dysfunctioning, schizophrenics with signs of brain dysfunctioning, depressed without signs of brain dysfunctioning, and depressed with signs of brain dysfunctioning. The Trail Making Test--Parts A & B, the difference between these two components of the Trail Making Test, the World Fluency Test and a Laterality Index based on the age scale scores of the WAIS significantly discriminated between patients with and without brain dysfunction. Using these variables and a Linear Discriminant Function Analysis, we found that 84.5% of our subjects could be correctly classified. Substantially fewer patients could be correctly classified with respect to their psychiatric diagnosis (i.e., 66.7% correct classification). In fact, only the PIQ from the WAIS-R showed significant differences between the depressed and schizophrenic subjects. Poor neuropsychological performance was interpreted as showing more than the behavioral disorganization associated with psychiatric states and was felt to be related to the presence of objective signs of brain dysfunctioning.

Research paper thumbnail of Fluctuation assisted diffusion through ion channels

Physica A: Statistical Mechanics and its Applications, 2007

We present a discrete Markovian model to describe the transport of potassium ions in the selectiv... more We present a discrete Markovian model to describe the transport of potassium ions in the selectivity filter of the KcsA K + channel. Recent work by MacKinnon et al. has documented that an array of two K + ions alternating in position with water molecules partake in single-file transport ...

Research paper thumbnail of Stability of photochemical mechanisms for photogalvanic cells

Journal of Photochemistry, 1980

A general analysis of the stability properties of photochemical mechanisms for photogalvanic cell... more A general analysis of the stability properties of photochemical mechanisms for photogalvanic cells is presented. We consider a model, recently introduced, for photoredox systems in which a one-electron redox couple (here an inorganic couple) reacts with a two-electron redox couple (here an organic dye). We consider first the case of a well-stirred cell. A linear stability analysis and then a thermodynamic stability analysis are carried out for each of six possible mechanisms allowed by the model; similar analyses are performed to determine the stability at certain coordinates of the kinetic "phase diagram" where two (or more) mechanisms may compete. Taken together, the results of these analyses indicate that the photostationary state is asymptotically stable in all the kinetic regimes investigated. We next study how these conclusions are changed when diffusion of the reactants is taken into account explicitly. The general analysis is then specialized to treat a particular well-known photogalvanic cell, the iron-thionine system. The identity and the stability of the photostationary state for this system are investigated and are then quantified as functions of the concentration of the metal ion Fe'+ and the system illumination. Our results on this specific photogalvanic system are examined in the light of known results on the stability properties of coupled chemical networks and are correlated with a recent study of the optimum efficiency of photogalvanic cells for solar energy conversion.

Research paper thumbnail of Lattice model of the early stages of the electrification of a cloud

Physical Review E, 2003

The early stages of the microphysics of the electrification process within a cloud are considered... more The early stages of the microphysics of the electrification process within a cloud are considered using a two-dimensional lattice model. Using insights generated from Monte Carlo simulations and the theory of finite Markov processes, the mean walk length statistics of the particles, the instantaneous electric potential and electric field profiles, the time evolution of electrostatic energy and their dependence on system size are studied. Some unexpected features of the kinetics of electrification and of the statistics of crossings of the threshold for an electric discharge to occur are brought out.

Research paper thumbnail of Chaos, Periodic Chaos, and the Random- Walk Problem

Physical Review Letters, 1982

The authors have studied whether numerically generated sequences from the logistic parabola f/sub... more The authors have studied whether numerically generated sequences from the logistic parabola f/sub b/(x) = 4bx(1-x) with b,xelement of(0,1), for values of b above the Feigenbaum critical value b/sub infinity/, are truly chaotic or whether they are periodic but with exceedingly large periods and very long transients. Using the logistic parabola the authors calculate via Monte Carlo simulation the average walk length for trapping on a one-dimensional lattice with a centrosymmetric trap. Comparison with exact results suggests that the only ''truly chaotic'' sequence is the one for which b = 1.

Research paper thumbnail of Influence of surface structure on the kinetics of diffusion-controlled reactive processes on molecular organizates and colloidal catalysts

The Journal of Physical Chemistry, 1989

Page 1. 3728 J. Phys. Chem. 1989, 93, 3728-3735 more compact, thereby resulting in lower compress... more Page 1. 3728 J. Phys. Chem. 1989, 93, 3728-3735 more compact, thereby resulting in lower compressibility. The adiabatic compressibility coefficient of aqueous micelle solutions of CTAB containing these solubilizates is seen ...

Research paper thumbnail of Modeling of Diffusion-Reaction Processes Involving Geminate Radical Pairs

The Journal of Physical Chemistry A, 2001

A Markovian theory is developed to study the efficiency of diffusion-reaction processes involving... more A Markovian theory is developed to study the efficiency of diffusion-reaction processes involving geminate radical pairs moving on the surface of a catalyst (e.g., zeolite) support. Results obtained for the simplest (lattice) system are corroborated by Monte Carlo calculations, and these are extended to study the effect of system size. A kinetic model is developed to account for the results obtained and leads to the general conclusion that the photochemical generation of a triplet pair, followed by the subsequent diffusion and eventual recombination of the radicals Rv and RV, lengthens the mean reaction time by a factor of ∼2, relative to the case in which Rv and RV are assumed to be present at the very outset.

Research paper thumbnail of Spectator-ion effect on the passage of ions through membranes

Proceedings of the National Academy of Sciences, 1975

In this paper, we investigate the interplay between geometric and dielectric factors in influenci... more In this paper, we investigate the interplay between geometric and dielectric factors in influencing the image force acting on an ion passing through a membrane, for a system having the approximate dimensions of Escherichia coli. We also study the effect of one ion in a membrane on the passage of a second ion through the membrane, by calculating the radial and angular forces experienced by the second ion due to the presence of the "spectator ion." Our conclusions follow from numerical studies on expressions obtained by solving (exactly) Laplace's equation for the model assumed in this paper. The conclusions are: (i) small changes in the dielectric constant of the membrane are far more significant in determining the image force acting on an ion in a membrane than dramatic changes in the dielectric character of the regions interior and/or exterior to the cell; (ii) a spectator ion in a membrane situated near a boundary may influence in a significant way the passage of a s...

Research paper thumbnail of Pattern development in cellular automata triggered by site-specific reactive processes

Physics Letters A, 1988

The (discretized) evolution of regular and fractal patterns, initiated by a site-specific event, ... more The (discretized) evolution of regular and fractal patterns, initiated by a site-specific event, is studied via simulation on a simple model. Consistent with recent theories of diffusion on percolation networks, our results show that the development of fractal patterns is distinctly slower than for regular (euclidean) patterns. The possible relevance. of our results to the evolution of symmetry-breaking instabilities is brought out.

Research paper thumbnail of Spectral dimension of regular and fractal lattices

Physics Letters A, 1990

The stochastic master equation dP 1/dt=->.~is solved for finite lattices with the boundary condit... more The stochastic master equation dP 1/dt=->.~is solved for finite lattices with the boundary condition P1(0) =ô,,,,, m being an interior site of the lattice. It is observed that the entropy function S(t) =-~P, (1) In[P,(t) J grows linearly with In (t). The slope is 1d~, where dĩs the spectral dimension of the lattice. Self-similar spaces are characterized by three quantities: de, the dimension of the embedding Eu-2'~~5 =-Pk in (Pk) un (n)

Research paper thumbnail of Critical Point Correlations of the Yvon-Born-Green Equation

Physical Review Letters, 1981

The critical behavior of the Yvon-Born-Green integral equation for fluids is analyzed by a moment... more The critical behavior of the Yvon-Born-Green integral equation for fluids is analyzed by a moment expansion which yields a nonlinear differential equation accurately describing the long-range correlations. Phase plane analyses show that for dimensions d «4 a critical point is characterized by p =4-d with g (r)-1 negative for large distances, x, in contrast to normal expectations. For d &4 the differential equation allows Ig(x)-1] &0 and p=0 or 4-d. The compressibility never diverges if 4=1.

Research paper thumbnail of Analytic expression for the mean time to absorption for a random walker on the Sierpinski fractal. III. The effect of non-nearest-neighbor jumps

Physical Review E, 2013

We present exact, analytic results for the mean time to trapping of a random walker on the class ... more We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d 2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearestneighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.

Research paper thumbnail of Aggregation of dipolar colloidal particles: Geometric effects

Physical Review E, 2006

To understand the importance of confinement and the influence of translational degrees of freedom... more To understand the importance of confinement and the influence of translational degrees of freedom on aggregation of dipolar colloidal particles, we calculate numerically-exact values for the mean encounter time for two nonspherically symmetric molecules to form a two-molecule cluster, regarded here as a precursor to aggregation. A lattice model is formulated in which the asymmetry of the molecules is accounted for by representing each as a "dimer" in the sense that each molecule is specified to occupy two adjacent lattice sites. The two dimers undergo simultaneous translation, and the mean times for their encounter are determined. Exact numerical results are obtained via application of the theory of finite Markov processes. The results allow one to examine in a detailed way the interplay among such factors as geometrical confinement, system size, translational motion, and specific orientational effects in influencing the aggregation event. The results are compared with previously reported theoretical predictions and experiments on the behavior of dipolar colloidal particles in the presence of an applied magnetic field.

Research paper thumbnail of Synchronous vs. asynchronous dynamics of diffusion-controlled reactions

Physica A: Statistical Mechanics and its Applications, 2003

An analytical method based on the classical ruin problem is developed to compute the mean reactio... more An analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice. At each time step, either both walkers diffuse simultaneously with probability p (synchronous event) or one of them diffuses while the other remains immobile with complementary probability (asynchronous event). Reaction takes place through same site occupation or position exchange. We study the influence of the degree of synchronicity p of the walkers and the lattice size N on the global reaction's efficiency. For odd N , the purely synchronous case (p = 1) is always the most effective one, while for even N , the encounter time is minimized by a combination of synchronous and asynchronous events. This new parity effect is fully confirmed by Monte Carlo simulations on 1d lattices as well as for 2d and 3d lattices. In contrast, the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of p. The relevance of the model for several research areas is briefly discussed.

Research paper thumbnail of Inverted regions induced by geometric constraints on a classical encounter-controlled binary reaction

Physica A: Statistical Mechanics and its Applications, 2006

The efficiency of an encounter-controlled reaction between two independently-mobile reactants on ... more The efficiency of an encounter-controlled reaction between two independently-mobile reactants on a lattice is characterized by the mean number n of steps to reaction. The two reactants are distinguished by their mass with the "light" walker performing a jump to a nearest-neighbor site in each time step, while the "heavy" walker hops only with a probability p; we associate p with the "temperature" of the system. To account for geometric exclusion effects in the reactive event, two reaction channels are specified for the walkers; irreversible reaction occurs either in a nearest-neighbor collision, or when the two reactants attempt to occupy the same site. Lattices subject to periodic and to confining boundary conditions are considered. For periodic lattices, depending on the initial state, the reaction time either falls off monotonically with p or displays a local minimum with respect to p; occurrence of the latter signals a regime where the efficiency of the reaction effectively decreases with increasing temperature. Such behavior can also occur when one averages over all initial conditions, but can disappear if the jump probability of the light walker falls below a characteristic threshold value. Even more robust behavior can occur on lattices subject to confining boundary conditions. Depending on the initial conditions, the reaction time as a function of p may increase monotonically, decrease monotonically, display a single maximum or even a maximum and minimum; in the latter case, one can identify distinct regimes where the above-noted inversion in reaction efficiency can occur. We document both numerically and theoretically that these inversion regions are a consequence of a strictly classical interplay between excluded volume effects implicit in the specification of the two reaction channels, and the system's dimensionality and spatial extent. Our results highlight situations where the description of an encounter-controlled reactive event cannot be described by a single, effective diffusion coefficient. We also distinguish between the inversion region identified here and the Marcus inverted region which arises in electron transfer reactions.

Research paper thumbnail of Statistical mechanics via the method of complementary variational principles

Molecular Physics, 1981

Page 1. MOLECULAR PHYSICS, 1981, VOL. 44, No. 3,761-781 Statistical mechanics via the method of c... more Page 1. MOLECULAR PHYSICS, 1981, VOL. 44, No. 3,761-781 Statistical mechanics via the method of complementary variational principles I. Application to classical, two dimensional fluids t by JOHN E. MAcCARTHY and JOHN ...

Research paper thumbnail of Maine South Basketball Varsity Roster 2012-2013 # Name Ht. Wt. Year Position

Research paper thumbnail of Comparative Study of Spontaneous Emission and Spin-Lattice Relaxation atT=0

Physical Review B, 1971

In this paper we present a comparative study of spontaneous emission and spin-lattice relaxation ... more In this paper we present a comparative study of spontaneous emission and spin-lattice relaxation at zero temperature. In particular, we study the time evolution of the density matrix for two simple models as determined from an analysis of the Prigogine-Rdsibois master equation. The first model treated is that of the Wigner-Weisskopf atom in a three-dimensional radiation field; the second model is that of a single, effective spin in interaction with the phonon modes of a three-dimensional lattice. The divergence which arises in the solution of the master equation for the first model is avoided using a frequency cutoff. A frequency cutoff in the second model is imposed by the upper bound of the spectrum of modes in the crystal, and this fact manifests itself when one integrates over the first Brillouin zone only. From a detailed numerical study of the analytic results obtained in solving the master equation, we find that for both models the relaxation to equilibrium is characterized, in part, by a sequence of slowly damped oscillations. This result seems to be in agreement with the observation made by Zwanzig, namely, that exponential decay in time seems not to be universal, and may, in fact, be hidden behind some other kind of time dependence. The numerical study also reveals, however, that the nonexponential modes of decay can be quantitatively different in magnitude and qualitatively different in structure for atomic versus spin systems. Finally, based on the solution obtained for the spin problem, an estimate is made of the relaxation time for cerium ethyl sulfate, and this estimate is found to be consistent with experiment.

Research paper thumbnail of Method for continuously casting a strip of alloyed metal

Research paper thumbnail of Critical Behavior of the Yvon-Born-Green Equation: Effects of Dimensionality

Physical Review Letters, 1982

Analytic studies of the Yvon-Born-Green equation have suggested that it may show no true critical... more Analytic studies of the Yvon-Born-Green equation have suggested that it may show no true critical region for spatial dimension d &4 but that for d & 4 there is a critical point of the mean-field type. Presented here are the results of numerical solution of the equation which strongly support these suggestions. Thus the Yvon-Born-Green equation exhibits the correct borderline dimensionality of four and the expected mean-field behavior for d 4, but for d & 4 has only a region of long, but finite, ranged correlations.

Research paper thumbnail of Neuropsychological Performance in Psychiatric Patients with or Without Documented Brain Dysfunction

International Journal of Neuroscience, 1988

The current study examined neuropsychological performance of schizophrenic and depressed patients... more The current study examined neuropsychological performance of schizophrenic and depressed patients with and without structural or EEG signs of brain dysfunctioning. The neuropsychological test battery was designed to sample intellectual functioning, psychomotor skills, nonverbal memory, and novel problem solving ability. Patients were classified into four groups: schizophrenics without signs of brain dysfunctioning, schizophrenics with signs of brain dysfunctioning, depressed without signs of brain dysfunctioning, and depressed with signs of brain dysfunctioning. The Trail Making Test--Parts A & B, the difference between these two components of the Trail Making Test, the World Fluency Test and a Laterality Index based on the age scale scores of the WAIS significantly discriminated between patients with and without brain dysfunction. Using these variables and a Linear Discriminant Function Analysis, we found that 84.5% of our subjects could be correctly classified. Substantially fewer patients could be correctly classified with respect to their psychiatric diagnosis (i.e., 66.7% correct classification). In fact, only the PIQ from the WAIS-R showed significant differences between the depressed and schizophrenic subjects. Poor neuropsychological performance was interpreted as showing more than the behavioral disorganization associated with psychiatric states and was felt to be related to the presence of objective signs of brain dysfunctioning.

Research paper thumbnail of Fluctuation assisted diffusion through ion channels

Physica A: Statistical Mechanics and its Applications, 2007

We present a discrete Markovian model to describe the transport of potassium ions in the selectiv... more We present a discrete Markovian model to describe the transport of potassium ions in the selectivity filter of the KcsA K + channel. Recent work by MacKinnon et al. has documented that an array of two K + ions alternating in position with water molecules partake in single-file transport ...

Research paper thumbnail of Stability of photochemical mechanisms for photogalvanic cells

Journal of Photochemistry, 1980

A general analysis of the stability properties of photochemical mechanisms for photogalvanic cell... more A general analysis of the stability properties of photochemical mechanisms for photogalvanic cells is presented. We consider a model, recently introduced, for photoredox systems in which a one-electron redox couple (here an inorganic couple) reacts with a two-electron redox couple (here an organic dye). We consider first the case of a well-stirred cell. A linear stability analysis and then a thermodynamic stability analysis are carried out for each of six possible mechanisms allowed by the model; similar analyses are performed to determine the stability at certain coordinates of the kinetic "phase diagram" where two (or more) mechanisms may compete. Taken together, the results of these analyses indicate that the photostationary state is asymptotically stable in all the kinetic regimes investigated. We next study how these conclusions are changed when diffusion of the reactants is taken into account explicitly. The general analysis is then specialized to treat a particular well-known photogalvanic cell, the iron-thionine system. The identity and the stability of the photostationary state for this system are investigated and are then quantified as functions of the concentration of the metal ion Fe'+ and the system illumination. Our results on this specific photogalvanic system are examined in the light of known results on the stability properties of coupled chemical networks and are correlated with a recent study of the optimum efficiency of photogalvanic cells for solar energy conversion.

Research paper thumbnail of Lattice model of the early stages of the electrification of a cloud

Physical Review E, 2003

The early stages of the microphysics of the electrification process within a cloud are considered... more The early stages of the microphysics of the electrification process within a cloud are considered using a two-dimensional lattice model. Using insights generated from Monte Carlo simulations and the theory of finite Markov processes, the mean walk length statistics of the particles, the instantaneous electric potential and electric field profiles, the time evolution of electrostatic energy and their dependence on system size are studied. Some unexpected features of the kinetics of electrification and of the statistics of crossings of the threshold for an electric discharge to occur are brought out.

Research paper thumbnail of Chaos, Periodic Chaos, and the Random- Walk Problem

Physical Review Letters, 1982

The authors have studied whether numerically generated sequences from the logistic parabola f/sub... more The authors have studied whether numerically generated sequences from the logistic parabola f/sub b/(x) = 4bx(1-x) with b,xelement of(0,1), for values of b above the Feigenbaum critical value b/sub infinity/, are truly chaotic or whether they are periodic but with exceedingly large periods and very long transients. Using the logistic parabola the authors calculate via Monte Carlo simulation the average walk length for trapping on a one-dimensional lattice with a centrosymmetric trap. Comparison with exact results suggests that the only ''truly chaotic'' sequence is the one for which b = 1.

Research paper thumbnail of Influence of surface structure on the kinetics of diffusion-controlled reactive processes on molecular organizates and colloidal catalysts

The Journal of Physical Chemistry, 1989

Page 1. 3728 J. Phys. Chem. 1989, 93, 3728-3735 more compact, thereby resulting in lower compress... more Page 1. 3728 J. Phys. Chem. 1989, 93, 3728-3735 more compact, thereby resulting in lower compressibility. The adiabatic compressibility coefficient of aqueous micelle solutions of CTAB containing these solubilizates is seen ...

Research paper thumbnail of Modeling of Diffusion-Reaction Processes Involving Geminate Radical Pairs

The Journal of Physical Chemistry A, 2001

A Markovian theory is developed to study the efficiency of diffusion-reaction processes involving... more A Markovian theory is developed to study the efficiency of diffusion-reaction processes involving geminate radical pairs moving on the surface of a catalyst (e.g., zeolite) support. Results obtained for the simplest (lattice) system are corroborated by Monte Carlo calculations, and these are extended to study the effect of system size. A kinetic model is developed to account for the results obtained and leads to the general conclusion that the photochemical generation of a triplet pair, followed by the subsequent diffusion and eventual recombination of the radicals Rv and RV, lengthens the mean reaction time by a factor of ∼2, relative to the case in which Rv and RV are assumed to be present at the very outset.

Research paper thumbnail of Spectator-ion effect on the passage of ions through membranes

Proceedings of the National Academy of Sciences, 1975

In this paper, we investigate the interplay between geometric and dielectric factors in influenci... more In this paper, we investigate the interplay between geometric and dielectric factors in influencing the image force acting on an ion passing through a membrane, for a system having the approximate dimensions of Escherichia coli. We also study the effect of one ion in a membrane on the passage of a second ion through the membrane, by calculating the radial and angular forces experienced by the second ion due to the presence of the "spectator ion." Our conclusions follow from numerical studies on expressions obtained by solving (exactly) Laplace's equation for the model assumed in this paper. The conclusions are: (i) small changes in the dielectric constant of the membrane are far more significant in determining the image force acting on an ion in a membrane than dramatic changes in the dielectric character of the regions interior and/or exterior to the cell; (ii) a spectator ion in a membrane situated near a boundary may influence in a significant way the passage of a s...

Research paper thumbnail of Pattern development in cellular automata triggered by site-specific reactive processes

Physics Letters A, 1988

The (discretized) evolution of regular and fractal patterns, initiated by a site-specific event, ... more The (discretized) evolution of regular and fractal patterns, initiated by a site-specific event, is studied via simulation on a simple model. Consistent with recent theories of diffusion on percolation networks, our results show that the development of fractal patterns is distinctly slower than for regular (euclidean) patterns. The possible relevance. of our results to the evolution of symmetry-breaking instabilities is brought out.

Research paper thumbnail of Spectral dimension of regular and fractal lattices

Physics Letters A, 1990

The stochastic master equation dP 1/dt=->.~is solved for finite lattices with the boundary condit... more The stochastic master equation dP 1/dt=->.~is solved for finite lattices with the boundary condition P1(0) =ô,,,,, m being an interior site of the lattice. It is observed that the entropy function S(t) =-~P, (1) In[P,(t) J grows linearly with In (t). The slope is 1d~, where dĩs the spectral dimension of the lattice. Self-similar spaces are characterized by three quantities: de, the dimension of the embedding Eu-2'~~5 =-Pk in (Pk) un (n)

Research paper thumbnail of Critical Point Correlations of the Yvon-Born-Green Equation

Physical Review Letters, 1981

The critical behavior of the Yvon-Born-Green integral equation for fluids is analyzed by a moment... more The critical behavior of the Yvon-Born-Green integral equation for fluids is analyzed by a moment expansion which yields a nonlinear differential equation accurately describing the long-range correlations. Phase plane analyses show that for dimensions d «4 a critical point is characterized by p =4-d with g (r)-1 negative for large distances, x, in contrast to normal expectations. For d &4 the differential equation allows Ig(x)-1] &0 and p=0 or 4-d. The compressibility never diverges if 4=1.

Research paper thumbnail of Analytic expression for the mean time to absorption for a random walker on the Sierpinski fractal. III. The effect of non-nearest-neighbor jumps

Physical Review E, 2013

We present exact, analytic results for the mean time to trapping of a random walker on the class ... more We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d 2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearestneighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.

Research paper thumbnail of Aggregation of dipolar colloidal particles: Geometric effects

Physical Review E, 2006

To understand the importance of confinement and the influence of translational degrees of freedom... more To understand the importance of confinement and the influence of translational degrees of freedom on aggregation of dipolar colloidal particles, we calculate numerically-exact values for the mean encounter time for two nonspherically symmetric molecules to form a two-molecule cluster, regarded here as a precursor to aggregation. A lattice model is formulated in which the asymmetry of the molecules is accounted for by representing each as a "dimer" in the sense that each molecule is specified to occupy two adjacent lattice sites. The two dimers undergo simultaneous translation, and the mean times for their encounter are determined. Exact numerical results are obtained via application of the theory of finite Markov processes. The results allow one to examine in a detailed way the interplay among such factors as geometrical confinement, system size, translational motion, and specific orientational effects in influencing the aggregation event. The results are compared with previously reported theoretical predictions and experiments on the behavior of dipolar colloidal particles in the presence of an applied magnetic field.

Research paper thumbnail of Synchronous vs. asynchronous dynamics of diffusion-controlled reactions

Physica A: Statistical Mechanics and its Applications, 2003

An analytical method based on the classical ruin problem is developed to compute the mean reactio... more An analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice. At each time step, either both walkers diffuse simultaneously with probability p (synchronous event) or one of them diffuses while the other remains immobile with complementary probability (asynchronous event). Reaction takes place through same site occupation or position exchange. We study the influence of the degree of synchronicity p of the walkers and the lattice size N on the global reaction's efficiency. For odd N , the purely synchronous case (p = 1) is always the most effective one, while for even N , the encounter time is minimized by a combination of synchronous and asynchronous events. This new parity effect is fully confirmed by Monte Carlo simulations on 1d lattices as well as for 2d and 3d lattices. In contrast, the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of p. The relevance of the model for several research areas is briefly discussed.

Research paper thumbnail of Inverted regions induced by geometric constraints on a classical encounter-controlled binary reaction

Physica A: Statistical Mechanics and its Applications, 2006

The efficiency of an encounter-controlled reaction between two independently-mobile reactants on ... more The efficiency of an encounter-controlled reaction between two independently-mobile reactants on a lattice is characterized by the mean number n of steps to reaction. The two reactants are distinguished by their mass with the "light" walker performing a jump to a nearest-neighbor site in each time step, while the "heavy" walker hops only with a probability p; we associate p with the "temperature" of the system. To account for geometric exclusion effects in the reactive event, two reaction channels are specified for the walkers; irreversible reaction occurs either in a nearest-neighbor collision, or when the two reactants attempt to occupy the same site. Lattices subject to periodic and to confining boundary conditions are considered. For periodic lattices, depending on the initial state, the reaction time either falls off monotonically with p or displays a local minimum with respect to p; occurrence of the latter signals a regime where the efficiency of the reaction effectively decreases with increasing temperature. Such behavior can also occur when one averages over all initial conditions, but can disappear if the jump probability of the light walker falls below a characteristic threshold value. Even more robust behavior can occur on lattices subject to confining boundary conditions. Depending on the initial conditions, the reaction time as a function of p may increase monotonically, decrease monotonically, display a single maximum or even a maximum and minimum; in the latter case, one can identify distinct regimes where the above-noted inversion in reaction efficiency can occur. We document both numerically and theoretically that these inversion regions are a consequence of a strictly classical interplay between excluded volume effects implicit in the specification of the two reaction channels, and the system's dimensionality and spatial extent. Our results highlight situations where the description of an encounter-controlled reactive event cannot be described by a single, effective diffusion coefficient. We also distinguish between the inversion region identified here and the Marcus inverted region which arises in electron transfer reactions.

Research paper thumbnail of Statistical mechanics via the method of complementary variational principles

Molecular Physics, 1981

Page 1. MOLECULAR PHYSICS, 1981, VOL. 44, No. 3,761-781 Statistical mechanics via the method of c... more Page 1. MOLECULAR PHYSICS, 1981, VOL. 44, No. 3,761-781 Statistical mechanics via the method of complementary variational principles I. Application to classical, two dimensional fluids t by JOHN E. MAcCARTHY and JOHN ...