scholarly journals Jump Markov models and transition state theory: the quasi-stationary distribution approach

2016 ◽  
Vol 195 ◽  
pp. 469-495 ◽  
Author(s):  
Giacomo Di Gesù ◽  
Tony Lelièvre ◽  
Dorian Le Peutrec ◽  
Boris Nectoux
Keyword(s):  
Markov Process ◽  
Transition State ◽  
State Space ◽  
Stationary Distribution ◽  
Transition State Theory ◽  
State Theory ◽  
Jump Markov Process ◽  
Continuous State Space ◽  
Continuous State ◽  
Quasi Stationary Distribution

We are interested in the connection between a metastable continuous state space Markov process (satisfyinge.g.the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring–Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring–Kramers formula to build kinetic Monte Carlo or Markov state models.

Convergence of quasi-stationary to stationary distributions for stochastically monotone Markov processes

1986 ◽  
Vol 23 (1) ◽  
pp. 215-220 ◽  
Author(s):  
Moshe Pollak ◽  
David Siegmund
Keyword(s):  
Markov Process ◽  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Stationary Distributions ◽  
New Process ◽  
Quasi Stationary Distribution

It is shown that if a stochastically monotone Markov process on [0,∞) with stationary distribution H has its state space truncated by making all states in [B,∞) absorbing, then the quasi-stationary distribution of the new process converges to H as B →∞.

Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state

1995 ◽  
Vol 27 (01) ◽  
pp. 120-145 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Process ◽  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Continuous Time ◽  
Hitting Time ◽  
Negative Integer ◽  
Absorbing State ◽  
Quasi Stationary Distribution

Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose P i (T < ∞) ≡ 1 and (*) lim i→∞ P i (T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff E i (e ∊T ) < ∞ for some ∊ > 0. The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.

Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state

1995 ◽  
Vol 27 (1) ◽  
pp. 120-145 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Process ◽  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Continuous Time ◽  
Hitting Time ◽  
Negative Integer ◽  
Absorbing State ◽  
Quasi Stationary Distribution

Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose Pi(T < ∞) ≡ 1 and (*) limi→∞Pi(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff Ei (e∊T) < ∞ for some ∊ > 0.The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.

Convergence of quasi-stationary to stationary distributions for stochastically monotone Markov processes

1986 ◽  
Vol 23 (01) ◽  
pp. 215-220 ◽  
Author(s):  
Moshe Pollak ◽  
David Siegmund
Keyword(s):  
Markov Process ◽  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Stationary Distributions ◽  
New Process ◽  
Quasi Stationary Distribution

It is shown that if a stochastically monotone Markov process on [0,∞) with stationary distribution H has its state space truncated by making all states in [B,∞) absorbing, then the quasi-stationary distribution of the new process converges to H as B →∞.

scholarly journals Exact aggregation of absorbing Markov processes using the quasi-stationary distribution

1994 ◽  
Vol 31 (3) ◽  
pp. 626-634 ◽  
Author(s):  
James Ledoux ◽  
Gerardo Rubino ◽  
Bruno Sericola
Keyword(s):  
Markov Process ◽  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Markovian Property ◽  
Starting Process ◽  
Initial Distributions ◽  
Quasi Stationary Distribution

We characterize the conditions under which an absorbing Markovian finite process (in discrete or continuous time) can be transformed into a new aggregated process conserving the Markovian property, whose states are elements of a given partition of the original state space. To obtain this characterization, a key tool is the quasi-stationary distribution associated with absorbing processes. It allows the absorbing case to be related to the irreducible one. We are able to calculate the set of all initial distributions of the starting process leading to an aggregated homogeneous Markov process by means of a finite algorithm. Finally, it is shown that the continuous-time case can always be reduced to the discrete one using the uniformization technique.

scholarly journals Serial dependence of a Markov process

1965 ◽  
Vol 5 (3) ◽  
pp. 299-314 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Time Series ◽  
Markov Process ◽  
State Space ◽  
Discrete Time ◽  
Borel Subset ◽  
Serial Dependence ◽  
State Spaces ◽  
Infinite Dimensional ◽  
Continuous State Space ◽  
Continuous State

Consider a Markov process defined in discrete time t = 1, 2, 3, hellip on a state space S. The state of the Process at time time t will be specifies by a random varable Vt, taking values in S. This paper presents some results concerning the behaviour of the saquence V1, V2, V3hellip, considered as a time series. In general, S will be assumed to be a Borel subset of an h-dimensional Euclideam space, where h is finite. The results apply, in particular, to a continuous state space, taking S to be an interval of the realine, or to discrete process having finitely or enumerably many states. Certain results, which are indicated in what follows, apply also to more general (infinite-dimensional) state spaces.

scholarly journals Exact aggregation of absorbing Markov processes using the quasi-stationary distribution

1994 ◽  
Vol 31 (03) ◽  
pp. 626-634 ◽  
Author(s):  
James Ledoux ◽  
Gerardo Rubino ◽  
Bruno Sericola
Keyword(s):  
Markov Process ◽  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Markovian Property ◽  
Starting Process ◽  
Initial Distributions ◽  
Quasi Stationary Distribution

We characterize the conditions under which an absorbing Markovian finite process (in discrete or continuous time) can be transformed into a new aggregated process conserving the Markovian property, whose states are elements of a given partition of the original state space. To obtain this characterization, a key tool is the quasi-stationary distribution associated with absorbing processes. It allows the absorbing case to be related to the irreducible one. We are able to calculate the set of all initial distributions of the starting process leading to an aggregated homogeneous Markov process by means of a finite algorithm. Finally, it is shown that the continuous-time case can always be reduced to the discrete one using the uniformization technique.

Microscopic Interpretation of Arrhenius Parameters

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Activation Energy ◽  
Transition State ◽  
Transition State Theory ◽  
Average Energy ◽  
Zero Point ◽  
State Theory ◽  
Exponential Factor ◽  
Quantum Tunnelling ◽  
Microscopic Interpretation ◽  
The Difference

This chapter reviews the microscopic interpretation of the pre-exponential factor and the activation energy in rate constant expressions of the Arrhenius form. The pre-exponential factor of apparent unimolecular reactions is, roughly, expected to be of the order of a vibrational frequency, whereas the pre-exponential factor of bimolecular reactions, roughly, is related to the number of collisions per unit time and per unit volume. The activation energy of an elementary reaction can be interpreted as the average energy of the molecules that react minus the average energy of the reactants. Specializing to conventional transition-state theory, the activation energy is related to the classical barrier height of the potential energy surface plus the difference in zero-point energies and average internal energies between the activated complex and the reactants. When quantum tunnelling is included in transition-state theory, the activation energy is reduced, compared to the interpretation given in conventional transition-state theory.

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