The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the... more The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by non-stationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and provide a generalization of Khinchin's theorem. Our work quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.
0 D ec 2 00 7 The Critical Exponent of the Fractional Langevin Equation is α c ≈ 0 . 402
We investigate the dynamical phase diagram of the fractional Langevin equation and show that crit... more We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent αc = 0.402±0.002 marks a transition to a non-monotonic under-damped phase. The critical exponent αR = 0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the under-damped, the over-damped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.
Journal of Physics A: Mathematical and Theoretical, 2021
Randomly repeated measurements during the evolution of a closed quantum system create a sequence ... more Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements until the first detection is an integer, namely the dimensionality of the accessible Hilbert space. Moreover, the mean first detected return time is equal to the average time step between successive measurements times the mean number of measurements. Thus, the mean first detected return time scales linearly with the dimensionality of the accessible Hilbert space. The main goal of this work is to explain the quantization of the mean return time in terms of a quantized Berry phase.
We calculate the power spectrum of the noisy current in a narrow channel that admits one charge c... more We calculate the power spectrum of the noisy current in a narrow channel that admits one charge carrier at a time from the left L or right R reservoirs. The random positive and negative currents are produced by the random arrival and departure of discrete charges at a measuring device on one side. The unidirectional currents are not independent because the presence of one charge in the channel blocks the entrance of another. Four classes of trajectories are possible: the trans trajectories LR and RL and cis trajectories LL and RR. The power spectrum of the total current is described by explicit formulas, depending only on the statistics of the interarrival times and times spent in the channel. These formulas generalize those of shot ͑e.g., Schottky͒ noise and predict more complex behavior than the sum of the noises of the four types of trajectories, if they were independent: ͑i͒ the mean current and the intensity of fluctuations saturate as the arrival time on one side decreases and ͑ii͒ the noise intensity depends nonlinearly on the mean net current. Explicit formulas are given and special cases are analyzed. ͓S1063-651X͑96͒03808-1͔
This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory descript... more This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau-Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very recent mathematical field of research. Considering a spatially extended version of the Pomeau-Manneville map leads us to the phenomenon of anomalous diffusion. This problem will be discussed by applying stochastic continuous time random walk theory and by deriving a fractional diffusion equation. Another important topic within modern nonequilibrium statistical physics are fluctuation relations, which we investigate for anomalous dynamics. The chapter concludes by showing the importance of anomalous dynamics for understanding experimental results on biological cell migration.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a ... more A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's Q parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g (2) (t). Using a semi-classical approach and linear response theory, we show that the Q parameter can be described by a three-time dipole correlation function. This approach generalizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime Q can be large even in the long time limit, while in the fast modulation regime it becomes small.
First-Passage Phenomena and Their Applications, 2014
Measurements of spatial diffusion of cold atoms in optical lattices have revealed anomalous super... more Measurements of spatial diffusion of cold atoms in optical lattices have revealed anomalous super-diffusion, which is controlled by the depth of the optical lattice. We use first passage time statistics to derive the diffusion front of the atoms. In particular, the distributions of areas swept under the first passage curve till its first arrival, and of areas under the Bessel excursion are shown to be powerful tools in the analysis of the atomic cloud. A rather general relation between first passage time statistics and diffusivity is discussed, showing that first passage time analysis is a useful tool in the calculation of transport coefficients. A brief introduction to the semi-classical description of Sisyphus cooling is provided which yields a rich phase diagram for the dynamics.
We derive backward and forward fractional Schrödinger type of equations for the distribution of f... more We derive backward and forward fractional Schrödinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL 96, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.
The power spectrum of quantum dot fluorescence exhibits 1/f β noise, related to the intermittency... more The power spectrum of quantum dot fluorescence exhibits 1/f β noise, related to the intermittency of these nanosystems. As in other systems exhibiting 1/f noise, this power spectrum is not integrable at low frequencies, which appears to imply infinite total power. We report measurements of individual quantum dots that address this long-standing paradox. We find that the level of 1/f β noise decays with the observation time. The change of the spectrum with time places a bound on the total power. These observations are in stark contrast with most measurements of noise in macroscopic systems which do not exhibit any evidence for non-stationarity. We show that the traditional description of the power spectrum with a single exponent β is incomplete and three additional critical exponents characterize the dependence on experimental time.
We introduce a Lévy-Lorentz gas in which a light particle is scattered by static point scatterers... more We introduce a Lévy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers ͕ i ͖ are independent random variables identically distributed according to the probability density function ()ϳ Ϫ(1ϩ␥). We show that under certain conditions the mean square displacement of the particle obeys ͗x 2 (t)͘уCt 3Ϫ␥ for 1Ͻ␥Ͻ2. This behavior is compatible with a renewal Lévy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process.
Journal of Physics A: Mathematical and Theoretical, 2014
We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) x 2 (t) ≃... more We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) x 2 (t) ≃ log γ t. Comparison of annealed continuous time random walks (CTRWs) with logarithmic waiting time distribution ψ(τ) ≃ 1/(τ log 1+γ τ) and Sinai diffusion in quenched random landscapes shows striking similarities, despite their very different physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time and ensemble averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble and time averaged MSD.
We introduce a fractional Kramers equation for a particle interacting with a thermal heat bath an... more We introduce a fractional Kramers equation for a particle interacting with a thermal heat bath and external nonlinear force field. For the force-free case, the velocity damping follows the Mittag-Leffler relaxation and the diffusion is enhanced. The equation obeys the generalized Einstein relation, and its stationary solution is the Boltzmann distribution. Our results are compared to previous results on enhanced Lévy type of diffusion derived from stochastic collision models.
We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean ... more We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean x eq for overdamped Brownian motion in a binding potential U (x). Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large non-ergodic fluctuations and show how they are related to a super-aging correlation function.
Pesin's identity provides a profound connection between the Kolmogorov-Sinai entropy h KS and the... more Pesin's identity provides a profound connection between the Kolmogorov-Sinai entropy h KS and the Lyapunov exponent. It is well known that many systems exhibit subexponential separation of nearby trajectories and then ¼ 0. In many cases such systems are nonergodic and do not obey usual statistical mechanics. Here we investigate the nonergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows x t ¼ x 0 e t with 0 <<1. The limit distribution of is the inverse Lévy function. The average h i is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.
Physica A: Statistical Mechanics and its Applications, 1998
We investigate the in uence of a weak uniform bias on chaotic di usion generated by iterated one-... more We investigate the in uence of a weak uniform bias on chaotic di usion generated by iterated one-dimensional maps which, in the absence of the bias, lead to subdi usion.
We investigate statistics of occupation times for an over-damped Brownian particle in an external... more We investigate statistics of occupation times for an over-damped Brownian particle in an external force field. A backward Fokker-Planck equation introduced by Majumdar and Comtet describing the distribution of occupation times is solved. The solution gives a general relation between occupation time statistics and probability currents which are found from solutions of the corresponding problem of first passage time. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker-Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding potential rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.
To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts... more To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, a stochastic collision model is investigated. We consider the dynamics of a tracer particle of mass M , undergoing elastic collisions with ideal gas particles of mass m, in the Rayleigh limit m << M. The probability density function (PDF) of the gas particle velocity is f (ṽ m). Assuming a uniform collision rate and molecular chaos, we obtain the equilibrium distribution for the velocity of the tracer particle W eq (V M). Depending on asymptotic properties of f (ṽ m) we find that W eq (V M) is either the Maxwell velocity distribution or a Lévy distribution. In particular our results yield a generalized Maxwell distribution based on Lévy statistics using two approaches. In the first a thermodynamic argument is used, imposing on the dynamics the condition that equilibrium properties of the heavy tracer particle be independent of the coupling ǫ = m/M to the gas particles, similar to what is found for a Brownian particle in a fluid. This approach leads to a generalized temperature concept. In the second approach it is assumed that bath particles velocity PDF scales with an energy scale, i.e. the (nearly) ordinary temperature, as found in standard statistical mechanics. The two approaches yield different types of Lévy equilibrium which merge into a unique solution only for the Maxwell-Boltzmann case. Thus, relation between thermodynamics and statistical mechanics becomes non-trivial for the power law case. Finally, the relation of the kinetic model to fractional Fokker-Planck equations is discussed.
The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the... more The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by non-stationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and provide a generalization of Khinchin's theorem. Our work quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.
0 D ec 2 00 7 The Critical Exponent of the Fractional Langevin Equation is α c ≈ 0 . 402
We investigate the dynamical phase diagram of the fractional Langevin equation and show that crit... more We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent αc = 0.402±0.002 marks a transition to a non-monotonic under-damped phase. The critical exponent αR = 0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the under-damped, the over-damped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.
Journal of Physics A: Mathematical and Theoretical, 2021
Randomly repeated measurements during the evolution of a closed quantum system create a sequence ... more Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements until the first detection is an integer, namely the dimensionality of the accessible Hilbert space. Moreover, the mean first detected return time is equal to the average time step between successive measurements times the mean number of measurements. Thus, the mean first detected return time scales linearly with the dimensionality of the accessible Hilbert space. The main goal of this work is to explain the quantization of the mean return time in terms of a quantized Berry phase.
We calculate the power spectrum of the noisy current in a narrow channel that admits one charge c... more We calculate the power spectrum of the noisy current in a narrow channel that admits one charge carrier at a time from the left L or right R reservoirs. The random positive and negative currents are produced by the random arrival and departure of discrete charges at a measuring device on one side. The unidirectional currents are not independent because the presence of one charge in the channel blocks the entrance of another. Four classes of trajectories are possible: the trans trajectories LR and RL and cis trajectories LL and RR. The power spectrum of the total current is described by explicit formulas, depending only on the statistics of the interarrival times and times spent in the channel. These formulas generalize those of shot ͑e.g., Schottky͒ noise and predict more complex behavior than the sum of the noises of the four types of trajectories, if they were independent: ͑i͒ the mean current and the intensity of fluctuations saturate as the arrival time on one side decreases and ͑ii͒ the noise intensity depends nonlinearly on the mean net current. Explicit formulas are given and special cases are analyzed. ͓S1063-651X͑96͒03808-1͔
This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory descript... more This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau-Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very recent mathematical field of research. Considering a spatially extended version of the Pomeau-Manneville map leads us to the phenomenon of anomalous diffusion. This problem will be discussed by applying stochastic continuous time random walk theory and by deriving a fractional diffusion equation. Another important topic within modern nonequilibrium statistical physics are fluctuation relations, which we investigate for anomalous dynamics. The chapter concludes by showing the importance of anomalous dynamics for understanding experimental results on biological cell migration.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a ... more A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's Q parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g (2) (t). Using a semi-classical approach and linear response theory, we show that the Q parameter can be described by a three-time dipole correlation function. This approach generalizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime Q can be large even in the long time limit, while in the fast modulation regime it becomes small.
First-Passage Phenomena and Their Applications, 2014
Measurements of spatial diffusion of cold atoms in optical lattices have revealed anomalous super... more Measurements of spatial diffusion of cold atoms in optical lattices have revealed anomalous super-diffusion, which is controlled by the depth of the optical lattice. We use first passage time statistics to derive the diffusion front of the atoms. In particular, the distributions of areas swept under the first passage curve till its first arrival, and of areas under the Bessel excursion are shown to be powerful tools in the analysis of the atomic cloud. A rather general relation between first passage time statistics and diffusivity is discussed, showing that first passage time analysis is a useful tool in the calculation of transport coefficients. A brief introduction to the semi-classical description of Sisyphus cooling is provided which yields a rich phase diagram for the dynamics.
We derive backward and forward fractional Schrödinger type of equations for the distribution of f... more We derive backward and forward fractional Schrödinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL 96, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.
The power spectrum of quantum dot fluorescence exhibits 1/f β noise, related to the intermittency... more The power spectrum of quantum dot fluorescence exhibits 1/f β noise, related to the intermittency of these nanosystems. As in other systems exhibiting 1/f noise, this power spectrum is not integrable at low frequencies, which appears to imply infinite total power. We report measurements of individual quantum dots that address this long-standing paradox. We find that the level of 1/f β noise decays with the observation time. The change of the spectrum with time places a bound on the total power. These observations are in stark contrast with most measurements of noise in macroscopic systems which do not exhibit any evidence for non-stationarity. We show that the traditional description of the power spectrum with a single exponent β is incomplete and three additional critical exponents characterize the dependence on experimental time.
We introduce a Lévy-Lorentz gas in which a light particle is scattered by static point scatterers... more We introduce a Lévy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers ͕ i ͖ are independent random variables identically distributed according to the probability density function ()ϳ Ϫ(1ϩ␥). We show that under certain conditions the mean square displacement of the particle obeys ͗x 2 (t)͘уCt 3Ϫ␥ for 1Ͻ␥Ͻ2. This behavior is compatible with a renewal Lévy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process.
Journal of Physics A: Mathematical and Theoretical, 2014
We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) x 2 (t) ≃... more We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) x 2 (t) ≃ log γ t. Comparison of annealed continuous time random walks (CTRWs) with logarithmic waiting time distribution ψ(τ) ≃ 1/(τ log 1+γ τ) and Sinai diffusion in quenched random landscapes shows striking similarities, despite their very different physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time and ensemble averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble and time averaged MSD.
We introduce a fractional Kramers equation for a particle interacting with a thermal heat bath an... more We introduce a fractional Kramers equation for a particle interacting with a thermal heat bath and external nonlinear force field. For the force-free case, the velocity damping follows the Mittag-Leffler relaxation and the diffusion is enhanced. The equation obeys the generalized Einstein relation, and its stationary solution is the Boltzmann distribution. Our results are compared to previous results on enhanced Lévy type of diffusion derived from stochastic collision models.
We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean ... more We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean x eq for overdamped Brownian motion in a binding potential U (x). Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large non-ergodic fluctuations and show how they are related to a super-aging correlation function.
Pesin's identity provides a profound connection between the Kolmogorov-Sinai entropy h KS and the... more Pesin's identity provides a profound connection between the Kolmogorov-Sinai entropy h KS and the Lyapunov exponent. It is well known that many systems exhibit subexponential separation of nearby trajectories and then ¼ 0. In many cases such systems are nonergodic and do not obey usual statistical mechanics. Here we investigate the nonergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows x t ¼ x 0 e t with 0 <<1. The limit distribution of is the inverse Lévy function. The average h i is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.
Physica A: Statistical Mechanics and its Applications, 1998
We investigate the in uence of a weak uniform bias on chaotic di usion generated by iterated one-... more We investigate the in uence of a weak uniform bias on chaotic di usion generated by iterated one-dimensional maps which, in the absence of the bias, lead to subdi usion.
We investigate statistics of occupation times for an over-damped Brownian particle in an external... more We investigate statistics of occupation times for an over-damped Brownian particle in an external force field. A backward Fokker-Planck equation introduced by Majumdar and Comtet describing the distribution of occupation times is solved. The solution gives a general relation between occupation time statistics and probability currents which are found from solutions of the corresponding problem of first passage time. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker-Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding potential rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.
To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts... more To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, a stochastic collision model is investigated. We consider the dynamics of a tracer particle of mass M , undergoing elastic collisions with ideal gas particles of mass m, in the Rayleigh limit m << M. The probability density function (PDF) of the gas particle velocity is f (ṽ m). Assuming a uniform collision rate and molecular chaos, we obtain the equilibrium distribution for the velocity of the tracer particle W eq (V M). Depending on asymptotic properties of f (ṽ m) we find that W eq (V M) is either the Maxwell velocity distribution or a Lévy distribution. In particular our results yield a generalized Maxwell distribution based on Lévy statistics using two approaches. In the first a thermodynamic argument is used, imposing on the dynamics the condition that equilibrium properties of the heavy tracer particle be independent of the coupling ǫ = m/M to the gas particles, similar to what is found for a Brownian particle in a fluid. This approach leads to a generalized temperature concept. In the second approach it is assumed that bath particles velocity PDF scales with an energy scale, i.e. the (nearly) ordinary temperature, as found in standard statistical mechanics. The two approaches yield different types of Lévy equilibrium which merge into a unique solution only for the Maxwell-Boltzmann case. Thus, relation between thermodynamics and statistical mechanics becomes non-trivial for the power law case. Finally, the relation of the kinetic model to fractional Fokker-Planck equations is discussed.
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Papers by E. Barkai