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### Ergodicity of one-dimensional systems coupled to the logistic thermostat

##### Diego Tapias, Alessandro Bravetti and *David P. Sanders*

Submitted for publication
We analyze the ergodicity of three one-dimensional Hamiltonian systems, with harmonic, quartic and Mexican-hat potentials, coupled to the logistic thermostat. As criteria for ergodicity we employ: the independence of the Lyapunov spectrum with respect to initial conditions; the absence of visual "holes" in two-dimensional Poincar\'e sections; the agreement between the histograms in each variable and the theoretical marginal distributions; and the convergence of the global joint distribution to the theoretical one, as measured by the Hellinger distance. Taking a large number of random initial conditions, for certain parameter values of the thermostat we find no indication of regular trajectories and show that the time distribution converges to the ensemble one for an arbitrarily long trajectory for all the systems considered. Our results thus provide a robust numerical indication that the logistic thermostat can serve as a single one-parameter thermostat for stiff one-dimensional systems.

[arXiv:1611.05090]

### Geometric integrator for simulations in the canonical ensemble

##### Diego Tapias, *David P. Sanders*, Alessandro Bravetti

*J. Chem. Phys.*

**145**(08), 084113 (2016)

We introduce a geometric integrator for molecular dynamics simulations of physical systems in the canonical ensemble that preserves the invariant distribution in equations arising from the density dynamics algorithm, with any possible type of thermostat. Our integrator thus constitutes a unified framework that allows the study and comparison of different thermostats and of their influence on the equilibrium and non-equilibrium (thermo-)dynamic properties of a system. To show the validity and the generality of the integrator, we implement it with a second-order, time-reversible method and apply it to the simulation of a Lennard-Jones system with three different thermostats, obtaining good conservation of the geometrical properties and recovering the expected thermodynamic results. Moreover, to show the advantage of our geometric integrator over a non-geometric one, we compare the results with those obtained by using the non-geometric Gear integrator, which is frequently used to perform simulations in the canonical ensemble. The non-geometric integrator induces a drift in the invariant quantity, while our integrator has no such drift, thus ensuring that the system is effectively sampling the correct ensemble.

Journal | [arXiv:1605.01654]

### Efficient algorithms for the periodic Lorentz gas in two and three dimensions

##### Atahualpa S. Kraemer, Nikolay Kryukov, *David P. Sanders*

*J. Phys. A: Math. Theor.*

**49**(02), 025001 (2016)

We present efficient algorithms to calculate trajectories for periodic Lorentz gases consisting of square lattices of circular obstacles in two dimensions, and simple cubic lattices of spheres in three dimensions; these become increasingly efficient as the radius of the obstacles tends to 0, the so-called Boltzmann–Grad limit. The 2D algorithm applies continued fractions to obtain the exact disc with which a particle will collide at each step, instead of using periodic boundary conditions as in the classical algorithm. The 3D version incorporates the 2D algorithm by projecting to the three coordinate planes. As an application, we calculate distributions of free path lengths close to the Boltzmann–Grad limit for certain Lorentz gases. We also show how the algorithms may be applied to deal with general crystal lattices.

Journal | [arXiv:1511.00236]

### Horizons and free path distributions in quasiperiodic Lorentz gases

##### Atahualpa S. Kraemer, Michael Schmiedeberg, *David P. Sanders*

*Phys. Rev. E*

**92**(05), 052131 (2015)

We study the structure of quasiperiodic Lorentz gases, i.e., particles bouncing elastically off fixed obstacles arranged in quasiperiodic lattices. By employing a construction to embed such structures into a higher-dimensional periodic hyperlattice, we give a simple and efficient algorithm for numerical simulation of the dynamics of these systems. This same construction shows that quasiperiodic Lorentz gases generically exhibit a regime with infinite horizon, that is, empty channels through which the particles move without colliding, when the obstacles are small enough; in this case, the distribution of free paths is asymptotically a power law with exponent −3, as expected from infinite-horizon periodic Lorentz gases. For the critical radius at which these channels disappear, however, a new regime with locally finite horizon arises, where this distribution has an unexpected exponent of −5, previously observed only in a Lorentz gas formed by superposing three incommensurable periodic lattices in the Boltzmann-Grad limit where the radius of the obstacles tends to zero.

Journal | [arXiv:1511.00340]

### Lévy walks on lattices as multi-state processes

##### Giampaolo Cristadoro, Thomas Gilbert, Marco Lenci, *David P. Sanders*

*J. Stat. Mech.*

**2015**(05), P05012 (2015)

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which a regime of superdiffusion may lurk underneath one of normal diffusion.

Journal | [arXiv:1501.05216]

### Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards

##### Giampaolo Cristadoro, Thomas Gilbert, Marco Lenci, *David P. Sanders*

*Phys. Rev. E*

**90**(05), 050102(R) (2014)

We study diffusion on a periodic billiard table with an infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a Lévy walk combining exponentially distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients, which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards

Journal | [arXiv:arXiv:1408.0349]

### Transport properties of Léy walks: an analysis in terms of multistate processes

##### Giampaolo Cristadoro, Thomas Gilbert, Marco Lenci, *David P. Sanders*

*Europhys. Lett.*

**108**(5), 50002 (2014)

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of Lévy walks on lattices. By including exponentially distributed waiting times separating the successive jump events of a walker, we are led to a description of such Lévy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.

Journal | [arXiv:1407.0227]

### Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

##### Giampaolo Cristadoro, Thomas Gilbert, Marco Lenci & *David P. Sanders*

*Phys. Rev. E*

**90**(02), 022106 (2014)

We perform numerical measurements of the moments of the position of a tracer particle in a two-dimensional periodic billiard model (Lorentz gas) with infinite corridors. This model is known to exhibit a weak form of superdiffusion, in the sense that there is a logarithmic correction to the linear growth in time of the mean-squared displacement. We show numerically that this expected asymptotic behavior is easily overwhelmed by the subleading linear growth throughout the time range accessible to numerical simulations. We compare our simulations to analytical results for the variance of the anomalously rescaled limiting normal distributions.

Journal | [arXiv:1405.0975]

### Sustained currents in coupled diffusive systems

##### Hernán Larralde and *David P. Sanders*

*J. Phys. A: Math. Theor.*

**47**(34), 345001 (2014)

Coupling two diffusive systems may give rise to a nonequilibrium stationary state (NESS) with a non-trivial persistent, circulating current. We study a simple example that is exactly soluble, consisting of random walkers with different biases towards a reflecting boundary, modelling, for example, Brownian particles with different charge states in an electric field. We obtain analytical expressions for the concentrations and currents in the NESS for this model, and exhibit the main features of the system by numerical simulation.

Journal | [arXiv:1401.5526]

### Embedding quasicrystals in a periodic cell: Dynamics in quasiperiodic structures

##### Atahualpa S. Kraemer and *David P. Sanders*

*Phys. Rev. Lett.*

**111**(12), 125501 (2013)

We introduce a construction to embed a quasiperiodic lattice of obstacles into a single unit cell of a higher-dimensional space, with periodic boundary conditions. This construction transparently shows the existence of channels in these systems,in which particles may travel without colliding, up to a critical obstacle radius. It provides a simple and efficient algorithm for numerical simulation of dynamics in quasiperiodic structures, as well as giving a natural notion of uniform distribution (measure) and averages. As an application, we simulate diffusion in a two-dimensional quasicrystal, finding three different regimes, in particular atypical weak super-diffusion in the presence of channels, and sub-diffusion when obstacles overlap.

Journal | Preprint | [arXiv:1206.1103]

### Zero density of open paths in the Lorentz mirror model for arbitrary mirror probability

##### Atahualpa S. Kraemer & *David P. Sanders*

*J. Stat. Phys.*

**156**(5), 908 (2014)

We show, incorporating results obtained from numerical simulations, that in the Lorentz mirror model, the density of open paths in any finite box tends to 0 as the box size tends to infinity, for any mirror probability.

Journal | [arXiv:1406.4796]

### Encounter times in overlapping domains: application to epidemic spread in a population of territorial animals

##### Luca Giuggioli, Sebastian Pérez-Becker and *David P. Sanders*

*Phys. Rev. Lett.*

**110**(5), 058103 (2013)

We develop an analytical method to calculate encounter times of two random walkers in one dimension when each individual is segregated in its own spatial domain and shares with its neighbor only a fraction of the available space, finding very good agreement with numerically-exact calculations. We model a population of susceptible and infected territorial individuals with this spatial arrangement, and which may transmit an epidemic when they meet. We apply the results on encounter times to determine analytically the macroscopic propagation speed of the epidemic as a function of the microscopic characteristics: the confining geometry, the animal diffusion constant, and the infection transmission probability.

Journal | Preprint | [arXiv:1207.2427]

### Diffusion coefficients for periodically induced multi-step persistent walks on regular lattices

##### Thomas Gilbert and *David P. Sanders*

*J. Phys. A*

**45**(25), 255003 (2012)

We present a generalization of our formalism for the computation of diffusion coefficients of multi-step persistent random walks on regular lattices to walks which include zero-displacement states. This situation is especially relevant to systems where tracer particles move across potential barriers as a result of the action of a periodic forcing whose period sets the timescale between transitions.

Journal | [arXiv:]

### Structure and evolution of strange attractors in non-elastic triangular billiards

##### Aubin Arroyo, Roberto Markarian and *David P. Sanders*

*Chaos*

**22**(2), 026107 (2012)

We study pinball billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls are non-elastic: the outgoing angle with the normal vector to the boundary is a uniform factor $\lambda < 1$ smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter $\lambda$ is varied. For $\lambda$ in the interval (0, 1/3)$, we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of $\lambda$ the billiard dynamics gives rise to nonaccessible regions in phase space. For $\lambda$ close to 1, the attractor splits into three transitive components, the basins of attraction of which have fractal basin boundaries.

Journal | Preprint | [arXiv:1112.1255]

### Chaos and stability in a two-parameter family of convex billiard tables

##### Péter Bálint, Miklós Halász, Jorge A. Hernández-Tahuilán and *David P. Sanders*

*Nonlinearity*

**24**(5), 1499 (2011)

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.

Journal | Local copy | Preprint | [arXiv:1009.3922]

### Stable and unstable regimes in higher-dimensional convex billiards with cylindrical shape

##### Thomas Gilbert and *David P. Sanders*

*New J. Phys.*

**13**(2), 023040 (2011)

We introduce a class of convex, higher-dimensional billiard models that generalize stadium billiards. These models correspond to the free motion of a point particle in a region bounded by cylinders cut by planes. They are motivated by models of particles interacting via a string-type mechanism, and confined by hard walls. The combination of these elements may give rise to a defocusing mechanism, similar to that in two dimensions, which allows large chaotic regions in phase space. The remaining part of phase space is associated with marginally stable behaviour. In fact periodic orbits in these systems generically come in continuous parametric families, associated with a pair of parabolic eigendirections: the periodic orbits are unstable in the presence of a defocusing mechanism, but are marginally stable otherwise. By performing stability analysis of families of periodic orbits at a nonlinear level, we establish the conditions under which families are nonlinearly stable or unstable. As a result, we identify regions in the parameter space of the models that admit nonlinearly stable oscillations in the form of whispering gallery modes. Where no families of periodic orbits are stable, the billiards are completely chaotic, i.e. the Lyapunov exponents of the billiard map are non-zero.

Journal | Preprint | [arXiv:1009.0337]

### Diffusive properties of persistent walks on cubic lattices with application to periodic Lorentz gases

##### Thomas Gilbert, Huu Chuong Nguyen and *David P. Sanders*

*J. Phys. A: Math. Theor.*

**44**(6), 065001 (2011)

We calculate the diffusion coefficients of persistent random walks on cubic and hypercubic lattices, where the direction of a walker at a given step depends on the memory of one or two previous steps. These results are then applied to study a billiard model, namely a three-dimensional periodic Lorentz gas. The geometry of the model is studied in order to find the regimes in which it exhibits normal diffusion. In this regime, we calculate numerically the transition probabilities between cells to compare the persistent random-walk approximation with simulation results for the diffusion coefficient.

Journal | Local copy | Preprint | [arXiv:1009.3922]

### Diffusion coefficients for multi-step persistent random walks on lattices

##### T. Gilbert and *David P. Sanders*

*J. Phys. A: Math. Theor.*

**43**(3), 035001 (2010)

We calculate the diffusion coefficients of persistent random walks on lattices, where the direction of a walker at a given step depends on the memory of a certain number of previous steps. In particular, we describe a simple method which enables us to obtain explicit expressions for the diffusion coefficients of walks with two-step memory on different classes of one-, two- and higher-dimensional lattices.

Journal | Local copy | Preprint | [arXiv:0908.1271]

### Persistence effects in deterministic diffusion

##### T. Gilbert and *David P. Sanders*

*Phys. Rev. E*

**80**(4), 041121 (2009)

In systems which exhibit deterministic diffusion, the gross parameter dependence of the diffusion coefficient can often be understood in terms of random walk models. Provided the decay of correlations is fast enough, one can ignore memory effects and approximate the diffusion coefficient according to dimensional arguments. By successively including the effects of one and two steps of memory on this approximation, we examine the effects of ``persistence'' on the diffusion coefficients of extended two-dimensional billiard tables and show how to properly account for these effects, using walks in which a particle undergoes jumps in different directions with probabilities that depend on where they came from.

Journal | Local copy | Preprint | [arXiv: 0908.0600]

### Exact encounter times for many random walkers on regular and complex networks

##### *David P. Sanders*

*Phys. Rev. E*

**80**(3), 036119 (2009)

The exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated, on both regular and complex networks. Analytical results are obtained both for independent walkers, where any number of walkers can occupy the same site, and for walkers with an exclusion interaction, when no site can contain more than one walker. These analytical results are then compared with numerical simulations, showing very good agreement.

Journal | Local copy | Preprint | [arXiv:0906.0810]

### Long-range correlations in a simple stochastic model of coupled transport

##### H. Larralde and *David P. Sanders*

*J. Phys. A: Math. Theor.*

**42**(33), 335002 (2009)

We study coupled transport in the nonequilibrium stationary state of a model consisting of independent random walkers, moving along a one-dimensional channel, which carry a conserved energy-like quantity, with density and temperature gradients imposed by reservoirs at the ends of the channel. In our model, walkers interact with other walkers at the same site by sharing energy at each time step, but the amount of energy carried does not affect the motion of the walkers. We find that already in this simple model long-range correlations arise in the nonequilibrium stationary state which are similar to those observed in more realistic models of coupled transport.

Journal | Local copy | Preprint | [arXiv:0903.3166]

### Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries

##### A. Arroyo, R. Markarian and *David P. Sanders*

*Nonlinearity*

**22**(7), 1499 (2009)

We study the dynamics of billiard models with a modified collision rule: the outgoing angle from a collision is a uniform contraction, by a factor λ, of the incident angle. These pinball billiards interpolate between a one-dimensional map when λ=0 and the classical Hamiltonian case of elastic collisions when λ=1. For all λ < 1, the dynamics is dissipative, and thus gives rise to attractors, which may be periodic or chaotic. Motivated by recent rigorous results of Markarian, Pujals and Sambarino, we numerically investigate and characterise the bifurcations of the resulting attractors as the contraction parameter is varied. Some billiards exhibit only periodic attractors, some only chaotic attractors, and others have coexistence of the two types.

Journal | Local copy | Preprint | [arXiv:0902.1563]

### Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

##### *David P. Sanders*

*Phys. Rev. E*

**78**(6), 060101R (2008)

We show, both heuristically and numerically, that three-dimensional periodic Lorentz gases -- clouds of particles scattering off crystalline arrays of hard spheres -- often exhibit normal diffusion, even when there are gaps through which particles can travel without ever colliding, i.e., when the system has an infinite horizon. This is the case provided that these gaps are not "too big", as measured by their dimension. The results are illustrated with simulations of a simple three-dimensional model having different types of diffusive regime, and are then extended to higher-dimensional billiard models, which include hard-sphere fluids.

Journal | Local copy | Preprint | [arXiv:0808.2235]

### How rare are diffusive rare events?

##### *David P. Sanders* and H. Larralde

*Europhys. Lett.*

**82**(4), 40005 (2008)

We study the time until first occurrence, the

*first-passage time*, of rare density fluctuations in diffusive systems. We approach the problem using a model consisting of many independent random walkers on a lattice. The existence of spatial correlations makes this problem analytically intractable. However, for a mean-field approximation in which the walkers can jump anywhere in the system, we obtain a simple asymptotic form for the mean first-passage time to have a given number

*k*of particles at a distinguished site. We show numerically, and argue heuristically, that for large enough

*k*, the mean-field results give a good approximation for first-passage times for systems with nearest-neighbour dynamics, especially for two and higher spatial dimensions. Finally, we show how the results change when density fluctuations anywhere in the system, rather than at a specific distinguished site, are considered.

Journal | Local copy | Preprint | [arXiv:0804.1165]

### Competitive nucleation and the Ostwald rule in a generalized Potts model with multiple metastable phases

##### *David P. Sanders*, H. Larralde and F. Leyvraz

*Phys. Rev. B.*

**75**(13), 132101 (2007)

We introduce a simple nearest-neighbor spin model with multiple metastable phases, the number and decay pathways of which are explicitly controlled by the parameters of the system. With this model we can construct, for example, a system which evolves through an arbitrarily long succession of metastable phases. We also construct systems in which different phases may nucleate competitively from a single initial phase. For such a system, we present a general method to extract from numerical simulations the individual nucleation rates of the nucleating phases. The results show that the Ostwald rule, which predicts which phase will nucleate, must be modified probabilistically when the new phases are almost equally stable. Finally, we show that the nucleation rate of a phase depends, among other things, on the number of other phases accessible from it.

Journal | Local copy | Preprint | [arXiv:0704.0472]

### Metastability in Markov processes

##### H. Larralde, F. Leyvraz and *David P. Sanders*

*J. Stat. Mech.*

**2006**(08), P08013 (2006)

We present a formalism for describing slowly decaying systems in the context of finite Markov chains obeying detailed balance. We show that phase space can be partitioned into approximately decoupled regions, in which one may introduce restricted Markov chains which are close to the original process but do not leave these regions. Within this context, we identify the conditions under which the decaying system can be considered to be in a metastable state. Furthermore, we show that such metastable states can be described in thermodynamic terms and define their free energy. This is accomplished, showing that the probability distribution describing the metastable state is indeed proportional to the equilibrium distribution, as is commonly assumed. We test the formalism numerically in the case of the two-dimensional kinetic Ising model, using the Wang-Landau algorithm to show this proportionality explicitly, and confirm that the proportionality constant is as derived in the theory. Finally, we extend the formalism to situations in which a system can have several metastable states.

Journal | Local copy | Preprint | [arXiv:cond-mat/0608439]

### Occurrence of normal and anomalous diffusion in polygonal billiard channels

##### *David P. Sanders* and H. Larralde

*Phys. Rev. E*

**73**(2), 026205 (2006)

From extensive numerical simulations, we find that periodic polygonal billiard channels with angles which are irrational multiples of pi generically exhibit normal diffusion (linear growth of the mean squared displacement) when they have a finite horizon, i.e., when no particle can travel arbitrarily far without colliding. For the infinite horizon case we present numerical tests showing that the mean squared displacement instead grows asymptotically as t ln t. When the unit cell contains accessible parallel scatterers, however, we always find anomalous super-diffusion, i.e., power-law growth with an exponent larger than 1. This behavior cannot be accounted for quantitatively by a simple continuous-time random walk model. Instead, we argue that anomalous diffusion correlates with the existence of families of propagating periodic orbits. Finally we show that when a configuration with parallel scatterers is approached there is a crossover from normal to anomalous diffusion, with the diffusion coefficient exhibiting a power-law divergence.

Journal | Local copy | Preprint | [arXiv:cond-mat/0510654]

### Fine structure of distributions and central limit theorem in diffusive billiards

##### *David P. Sanders*

*Phys. Rev. E*

**71**(1), 016220 (2005)

We investigate deterministic diffusion in periodic billiard models, in terms of the convergence of rescaled distributions to the limiting normal distribution required by the central limit theorem; this is a stronger statement than the usual requirement that the mean square displacement grow asymptotically linearly in time. The main model studied is a chaotic Lorentz gas, where the central limit theorem has been rigorously proved. We study one-dimensional position and displacement densities describing the time evolution of statistical ensembles in a channel geometry, using a method which is more refined than histograms. We find a pronounced oscillatory fine structure, and show that this has its origin in the geometry of the billiard domain. This fine structure prevents the rescaled densities from converging pointwise to gaussian densities; however, demodulating them by the fine structure gives new densities which seem to converge uniformly. We give an analytical estimate of the rate of convergence of the original distributions to the limiting normal distribution based on the analysis of the fine structure, which agrees well with simulation results. We show that using a Maxwellian (gaussian) distribution of velocities in place of unit speed velocities does not affect the growth of the mean square displacement, but changes the limiting shape of the distributions to a non-gaussian one. Using the same methods, we give numerical evidence that a non-chaotic polygonal channel model also obeys the central limit theorem, but with a significantly slower rate of convergence.

Journal | Local copy | Preprint | [arXiv:nlin.CD/0411012]

### Chaos in cylindrical stadium billiards via a generic nonlinear mechanism

##### Thomas Gilbert and *David P. Sanders*

*Int. J. Bif. Chaos*

**22**(09), 1250206 (2012)

We describe conditions under which higher-dimensional billiard models in bounded, convex regions are fully chaotic, generalizing the Bunimovich stadium to dimensions above two. An example is a three-dimensional stadium bounded by a cylinder and several planes; the combination of these elements may give rise to defocusing, allowing large chaotic regions in phase space. By studying families of marginally-stable periodic orbits that populate the residual part of phase space, we identify conditions under which a nonlinear instability mechanism arises in their vicinity. For particular geometries, this mechanism rather induces stable nonlinear oscillations, including in the form of whispering-gallery modes.

Journal | Preprint | [arXiv:0908.4243]

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David P. Sanders Last modified: 13 December 2016