Peter Talkner - Academia.edu (original) (raw)
Papers by Peter Talkner
Physical Review E, Jan 1, 2004
Physica A: Statistical Mechanics and its Applications, 1997
The relation between mean first passage times T and transition rates F in noisy dynamical systems... more The relation between mean first passage times T and transition rates F in noisy dynamical systems with metastahle states is investigated. It is shown that the inverse mean first passage time to the separatrix of the noiseless system may deviate from twice the rate not only because in general the deterministic separatrix is not the locus in the state space from which a noisy trajectory goes to either side with equal probability. A further cause of a deviation from the often assumed relation FT = 1/2 between rates and mean first passage times is given if the noisy dynamics is discontinuous, i.e. shows jumps with finite probability. Then the value of the splitting probability at the separatrix does not fix the value of FT since the system need not visit the separatrix during a transition from one to the other side. Most important, for discontinuous processes the deviation from the FT = ½ rule survives even in the weak noise limit. A mathematical relation for the product of the rate and the mean first passage time is proposed for Markovian processes and numerically confirmed for a particular one-dimensional noisy map.
The Journal of chemical physics, Jan 7, 2014
Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized ... more Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized asymptotic Einstein relation is derived. It agrees with the well-known Einstein relation in the case of normal diffusion but continues to hold for sub- and super-diffusive spreading of the Brownian particle's mean square displacement. The generalized asymptotic Einstein relation is used to analyze data obtained from molecular dynamics simulations of a two-dimensional soft disk fluid. We mainly concentrated on medium densities for which we found super-diffusive behavior of a tagged fluid particle. At higher densities a range of normal diffusion can be identified. The motion presumably changes to sub-diffusion for even higher densities.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2007
The characteristic function of the work performed by an external time-dependent force on a Hamilt... more The characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system is identified with the time-ordered correlation function of the exponentiated system's Hamiltonian. A similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the Jarzynski work theorem.
The reduction of a continuous Markov process with multiple metastable states to a discrete rate p... more The reduction of a continuous Markov process with multiple metastable states to a discrete rate process is investigated in the presence of slow time dependent parameters such as periodic external forces or slowly fluctuating barrier heights. A quantitative criterion is provided under which condition a kinetic description with time dependent frozen rates applies. Finally it is shown how the long time behavior of the underlying continuous process can be retrieved from the knowledge of the discrete process by means of an appropriate random decoration of the discrete states. As a particular example of the presented theory an over-damped bistable Brownian oscillator with periodic driving is discussed.
Systems where resource availability approaches a critical threshold are common to many engineerin... more Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of
Physical Review E, 2008
Previously derived expressions for the characteristic function of work performed on a quantum sys... more Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the respective canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians. From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2014
Generalized measurements of an observable performed on a quantum system during a force protocol a... more Generalized measurements of an observable performed on a quantum system during a force protocol are investigated and conditions that guarantee the validity of the Jarzynski equality and the Crooks relation are formulated. In agreement with previous studies by M. Campisi, P. Talkner, and P. Hänggi [Phys. Rev. Lett. 105, 140601 (2010); Phys. Rev. E 83, 041114 (2011)], we find that these fluctuation relations are satisfied for projective measurements; however, for generalized measurements special conditions on the operators determining the measurements need to be met. For the Jarzynski equality to hold, the measurement operators of the forward protocol must be normalized in a particular way. The Crooks relation additionally entails that the backward and forward measurement operators depend on each other. Yet, quite some freedom is left as to how the two sets of operators are interrelated. This ambiguity is removed if one considers selective measurements, which are specified by a joint ...
Physical review. E, Statistical, nonlinear, and soft matter physics, 2014
Determining the work which is supplied to a system by an external agent provides a crucial step i... more Determining the work which is supplied to a system by an external agent provides a crucial step in any experimental realization of transient fluctuation relations. This, however, poses a problem for quantum systems, where the standard procedure requires the projective measurement of energy at the beginning and the end of the protocol. Unfortunately, projective measurements, which are preferable from the point of view of theory, seem to be difficult to implement experimentally. We demonstrate that, when using a particular type of generalized energy measurements, the resulting work statistics is simply related to that of projective measurements. This relation between the two work statistics entails the existence of modified transient fluctuation relations. The modifications are exclusively determined by the errors incurred in the generalized energy measurements. They are universal in the sense that they do not depend on the force protocol. Particularly simple expressions for the modif...
We present an approximate analytical expression for escape rates of time-dependent driven stochas... more We present an approximate analytical expression for escape rates of time-dependent driven stochastic processes with an absorbing boundary such as the driven leaky integrate-and-fire model for neural spiking. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. It is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The scheme is carefully tested and yields excellent agreement with three different numerical methods based on the Langevin equation, the Fokker-Planck equation and an integral equation. r 2005 Elsevier B.V. All rights reserved.
Reviews of Modern Physics, 1990
Reaction-rate theory: fifty years after Kramers. Peter Hänggi Lehrstuhl für Theoretische Physik, ... more Reaction-rate theory: fifty years after Kramers. Peter Hänggi Lehrstuhl für Theoretische Physik, University of Augsburg, D-8900 Augsburg, Federal Republic of Germany. Peter Talkner * Department of Physics, University of Basel, CH-4056 Basel, Switzerland. ...
Physical Review Letters, 2007
There is an intense debate in the recent literature about the correct generalization of Maxwell's... more There is an intense debate in the recent literature about the correct generalization of Maxwell's velocity distribution in special relativity. The most frequently discussed candidate distributions include the Jüttner function as well as modifications thereof. Here we report results from fully relativistic one-dimensional molecular dynamics simulations that resolve the ambiguity. The numerical evidence unequivocally favors the Jüttner distribution. Moreover, our simulations illustrate that the concept of ''thermal equilibrium'' extends naturally to special relativity only if a many-particle system is spatially confined. They make evident that ''temperature'' can be statistically defined and measured in an observer frame independent way.
Physical Review E, 2004
With this work we investigate an often neglected aspect of Brownian motor transport: The rôle of ... more With this work we investigate an often neglected aspect of Brownian motor transport: The rôle of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise. In doing so, we consider a Brownian inertial motor that is driven by an unbiased monochromatic, time-periodic force and thermal noise. Typically, we find that the asymptotic, time-and noiseaveraged transport velocities are small, possessing rather broad velocity fluctuations. This implies a corresponding poor performance for the rectification power. However, for tailored profiles of the ratchet potential and appropriate drive parameters, we can identify a drastic enhancement of the rectification efficiency. This regime is marked by persistent, uni-directional motion of the Brownian motor with few back-turns, only. The corresponding asymmetric velocity distribution is then rather narrow, with a support that predominantly favors only one sign for the velocity.
Physical Review E, 2005
We investigate anticipated synchronization between two periodically driven deterministic, dissipa... more We investigate anticipated synchronization between two periodically driven deterministic, dissipative inertia ratchets that are able to exhibit directed transport with a finite velocity. The two ratchets interact through an unidirectional delay coupling: one is acting as a master system while the other one represents the slave system. Each of the two dissipative deterministic ratchets is driven externally by a common periodic force. The delay coupling involves two parameters: the coupling strength and the (positive-valued) delay time. We study the synchronization features for the unbounded, current carrying trajectories of the master and the slave, respectively, for four different strengths of the driving amplitude. These in turn characterize differing phase space dynamics of the transporting ratchet dynamics: regular, intermittent and a chaotic transport regime.
Physical Review E, 2008
Various aspects of the statistics of work performed by an external classical force on a quantum m... more Various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator. In this special case two parameters are introduced that are sufficient to completely characterize the force protocol. Explicit results for the characteristic function of work and the respective probability distribution are provided and discussed for three different types of initial states of the oscillator: microcanonical, canonical and coherent states. Depending on the choice of the initial state the probability distributions of the performed work may grossly differ. This result in particular holds also true for identical force protocols. General fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed.
Chemical Physics, 2010
The reduction of a continuous Markov process with multiple metastable states to a discrete rate p... more The reduction of a continuous Markov process with multiple metastable states to a discrete rate process is investigated in the presence of slow time dependent parameters such as periodic external forces or slowly fluctuating barrier heights. A quantitative criterion is provided under which condition a kinetic description with time dependent frozen rates applies. Finally it is shown how the long time behavior of the underlying continuous process can be retrieved from the knowledge of the discrete process by means of an appropriate random decoration of the discrete states. As a particular example of the presented theory an over-damped bistable Brownian oscillator with periodic driving is discussed.
Chemical Physics, 2010
Brownian motion of single particles with various masses M and diameters D is studied by molecular... more Brownian motion of single particles with various masses M and diameters D is studied by molecular dynamics simulations. Besides the momentum auto-correlation function of the Brownian particle the memory function and the fluctuating force which enter the generalized Langevin equation of the Brownian particle are determined and their dependence on mass and diameter are investigated for two different fluid densities. Deviations of the fluctuating force distribution from a Gaussian form are observed for small particle diameters. For heavy particles the deviations of the fluctuating force from the total force acting on the Brownian particle decrease linearly with the mass ratio m/M where m denotes the mass of a fluid particle. (Eok Kyun Lee ) 1 J. Ingen-Housz was a dutch scholar and famous physician of his time who vaccinated the royal families of king George III and of empress Maria Theresia against smallpox. He was the first to recognize the importance of chlorophyll, the green matter (grüne Materie), as he called it, for the production of oxygen by plants.
Physical Review E, 2009
The evaluation of the specific heat of an open, damped quantum system is a subtle issue. One poss... more The evaluation of the specific heat of an open, damped quantum system is a subtle issue. One possible route is based on the thermodynamic partition function which is the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. Being an open system this quantity then naturally must be interpreted as the change of the specific heat obtained as the difference between the specific heat of the heat bath coupled to the system degrees of freedom and the specific heat of the bath alone. While this difference may become negative, the involved specific heats themselves are always positive; thus, the known thermodynamic stability criteria are perfectly guaranteed. For a damped quantum harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Stylized minimal models containing a single oscillator heat bath are employed to elucidate the occurrence of the anomalous temperature dependence of the corresponding specific heat values. Moreover, we comment on the consequences for the interpretation of the density of states based on the thermal partition function.
Physical Review E, Jan 1, 2004
Physica A: Statistical Mechanics and its Applications, 1997
The relation between mean first passage times T and transition rates F in noisy dynamical systems... more The relation between mean first passage times T and transition rates F in noisy dynamical systems with metastahle states is investigated. It is shown that the inverse mean first passage time to the separatrix of the noiseless system may deviate from twice the rate not only because in general the deterministic separatrix is not the locus in the state space from which a noisy trajectory goes to either side with equal probability. A further cause of a deviation from the often assumed relation FT = 1/2 between rates and mean first passage times is given if the noisy dynamics is discontinuous, i.e. shows jumps with finite probability. Then the value of the splitting probability at the separatrix does not fix the value of FT since the system need not visit the separatrix during a transition from one to the other side. Most important, for discontinuous processes the deviation from the FT = ½ rule survives even in the weak noise limit. A mathematical relation for the product of the rate and the mean first passage time is proposed for Markovian processes and numerically confirmed for a particular one-dimensional noisy map.
The Journal of chemical physics, Jan 7, 2014
Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized ... more Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized asymptotic Einstein relation is derived. It agrees with the well-known Einstein relation in the case of normal diffusion but continues to hold for sub- and super-diffusive spreading of the Brownian particle's mean square displacement. The generalized asymptotic Einstein relation is used to analyze data obtained from molecular dynamics simulations of a two-dimensional soft disk fluid. We mainly concentrated on medium densities for which we found super-diffusive behavior of a tagged fluid particle. At higher densities a range of normal diffusion can be identified. The motion presumably changes to sub-diffusion for even higher densities.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2007
The characteristic function of the work performed by an external time-dependent force on a Hamilt... more The characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system is identified with the time-ordered correlation function of the exponentiated system's Hamiltonian. A similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the Jarzynski work theorem.
The reduction of a continuous Markov process with multiple metastable states to a discrete rate p... more The reduction of a continuous Markov process with multiple metastable states to a discrete rate process is investigated in the presence of slow time dependent parameters such as periodic external forces or slowly fluctuating barrier heights. A quantitative criterion is provided under which condition a kinetic description with time dependent frozen rates applies. Finally it is shown how the long time behavior of the underlying continuous process can be retrieved from the knowledge of the discrete process by means of an appropriate random decoration of the discrete states. As a particular example of the presented theory an over-damped bistable Brownian oscillator with periodic driving is discussed.
Systems where resource availability approaches a critical threshold are common to many engineerin... more Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of
Physical Review E, 2008
Previously derived expressions for the characteristic function of work performed on a quantum sys... more Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the respective canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians. From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2014
Generalized measurements of an observable performed on a quantum system during a force protocol a... more Generalized measurements of an observable performed on a quantum system during a force protocol are investigated and conditions that guarantee the validity of the Jarzynski equality and the Crooks relation are formulated. In agreement with previous studies by M. Campisi, P. Talkner, and P. Hänggi [Phys. Rev. Lett. 105, 140601 (2010); Phys. Rev. E 83, 041114 (2011)], we find that these fluctuation relations are satisfied for projective measurements; however, for generalized measurements special conditions on the operators determining the measurements need to be met. For the Jarzynski equality to hold, the measurement operators of the forward protocol must be normalized in a particular way. The Crooks relation additionally entails that the backward and forward measurement operators depend on each other. Yet, quite some freedom is left as to how the two sets of operators are interrelated. This ambiguity is removed if one considers selective measurements, which are specified by a joint ...
Physical review. E, Statistical, nonlinear, and soft matter physics, 2014
Determining the work which is supplied to a system by an external agent provides a crucial step i... more Determining the work which is supplied to a system by an external agent provides a crucial step in any experimental realization of transient fluctuation relations. This, however, poses a problem for quantum systems, where the standard procedure requires the projective measurement of energy at the beginning and the end of the protocol. Unfortunately, projective measurements, which are preferable from the point of view of theory, seem to be difficult to implement experimentally. We demonstrate that, when using a particular type of generalized energy measurements, the resulting work statistics is simply related to that of projective measurements. This relation between the two work statistics entails the existence of modified transient fluctuation relations. The modifications are exclusively determined by the errors incurred in the generalized energy measurements. They are universal in the sense that they do not depend on the force protocol. Particularly simple expressions for the modif...
We present an approximate analytical expression for escape rates of time-dependent driven stochas... more We present an approximate analytical expression for escape rates of time-dependent driven stochastic processes with an absorbing boundary such as the driven leaky integrate-and-fire model for neural spiking. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. It is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The scheme is carefully tested and yields excellent agreement with three different numerical methods based on the Langevin equation, the Fokker-Planck equation and an integral equation. r 2005 Elsevier B.V. All rights reserved.
Reviews of Modern Physics, 1990
Reaction-rate theory: fifty years after Kramers. Peter Hänggi Lehrstuhl für Theoretische Physik, ... more Reaction-rate theory: fifty years after Kramers. Peter Hänggi Lehrstuhl für Theoretische Physik, University of Augsburg, D-8900 Augsburg, Federal Republic of Germany. Peter Talkner * Department of Physics, University of Basel, CH-4056 Basel, Switzerland. ...
Physical Review Letters, 2007
There is an intense debate in the recent literature about the correct generalization of Maxwell's... more There is an intense debate in the recent literature about the correct generalization of Maxwell's velocity distribution in special relativity. The most frequently discussed candidate distributions include the Jüttner function as well as modifications thereof. Here we report results from fully relativistic one-dimensional molecular dynamics simulations that resolve the ambiguity. The numerical evidence unequivocally favors the Jüttner distribution. Moreover, our simulations illustrate that the concept of ''thermal equilibrium'' extends naturally to special relativity only if a many-particle system is spatially confined. They make evident that ''temperature'' can be statistically defined and measured in an observer frame independent way.
Physical Review E, 2004
With this work we investigate an often neglected aspect of Brownian motor transport: The rôle of ... more With this work we investigate an often neglected aspect of Brownian motor transport: The rôle of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise. In doing so, we consider a Brownian inertial motor that is driven by an unbiased monochromatic, time-periodic force and thermal noise. Typically, we find that the asymptotic, time-and noiseaveraged transport velocities are small, possessing rather broad velocity fluctuations. This implies a corresponding poor performance for the rectification power. However, for tailored profiles of the ratchet potential and appropriate drive parameters, we can identify a drastic enhancement of the rectification efficiency. This regime is marked by persistent, uni-directional motion of the Brownian motor with few back-turns, only. The corresponding asymmetric velocity distribution is then rather narrow, with a support that predominantly favors only one sign for the velocity.
Physical Review E, 2005
We investigate anticipated synchronization between two periodically driven deterministic, dissipa... more We investigate anticipated synchronization between two periodically driven deterministic, dissipative inertia ratchets that are able to exhibit directed transport with a finite velocity. The two ratchets interact through an unidirectional delay coupling: one is acting as a master system while the other one represents the slave system. Each of the two dissipative deterministic ratchets is driven externally by a common periodic force. The delay coupling involves two parameters: the coupling strength and the (positive-valued) delay time. We study the synchronization features for the unbounded, current carrying trajectories of the master and the slave, respectively, for four different strengths of the driving amplitude. These in turn characterize differing phase space dynamics of the transporting ratchet dynamics: regular, intermittent and a chaotic transport regime.
Physical Review E, 2008
Various aspects of the statistics of work performed by an external classical force on a quantum m... more Various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator. In this special case two parameters are introduced that are sufficient to completely characterize the force protocol. Explicit results for the characteristic function of work and the respective probability distribution are provided and discussed for three different types of initial states of the oscillator: microcanonical, canonical and coherent states. Depending on the choice of the initial state the probability distributions of the performed work may grossly differ. This result in particular holds also true for identical force protocols. General fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed.
Chemical Physics, 2010
The reduction of a continuous Markov process with multiple metastable states to a discrete rate p... more The reduction of a continuous Markov process with multiple metastable states to a discrete rate process is investigated in the presence of slow time dependent parameters such as periodic external forces or slowly fluctuating barrier heights. A quantitative criterion is provided under which condition a kinetic description with time dependent frozen rates applies. Finally it is shown how the long time behavior of the underlying continuous process can be retrieved from the knowledge of the discrete process by means of an appropriate random decoration of the discrete states. As a particular example of the presented theory an over-damped bistable Brownian oscillator with periodic driving is discussed.
Chemical Physics, 2010
Brownian motion of single particles with various masses M and diameters D is studied by molecular... more Brownian motion of single particles with various masses M and diameters D is studied by molecular dynamics simulations. Besides the momentum auto-correlation function of the Brownian particle the memory function and the fluctuating force which enter the generalized Langevin equation of the Brownian particle are determined and their dependence on mass and diameter are investigated for two different fluid densities. Deviations of the fluctuating force distribution from a Gaussian form are observed for small particle diameters. For heavy particles the deviations of the fluctuating force from the total force acting on the Brownian particle decrease linearly with the mass ratio m/M where m denotes the mass of a fluid particle. (Eok Kyun Lee ) 1 J. Ingen-Housz was a dutch scholar and famous physician of his time who vaccinated the royal families of king George III and of empress Maria Theresia against smallpox. He was the first to recognize the importance of chlorophyll, the green matter (grüne Materie), as he called it, for the production of oxygen by plants.
Physical Review E, 2009
The evaluation of the specific heat of an open, damped quantum system is a subtle issue. One poss... more The evaluation of the specific heat of an open, damped quantum system is a subtle issue. One possible route is based on the thermodynamic partition function which is the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. Being an open system this quantity then naturally must be interpreted as the change of the specific heat obtained as the difference between the specific heat of the heat bath coupled to the system degrees of freedom and the specific heat of the bath alone. While this difference may become negative, the involved specific heats themselves are always positive; thus, the known thermodynamic stability criteria are perfectly guaranteed. For a damped quantum harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Stylized minimal models containing a single oscillator heat bath are employed to elucidate the occurrence of the anomalous temperature dependence of the corresponding specific heat values. Moreover, we comment on the consequences for the interpretation of the density of states based on the thermal partition function.