Heat Release by Controlled Continuous-Time Markov Jump Processes

Muratore-Ginanneschi, Paolo; Mejía‐Monasterio, Carlos; Peliti, Luca

doi:10.1007/s10955-012-0676-6

Cited by 22 publications

(41 citation statements)

References 54 publications

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“…For example this happens when only nearest-neighbor jumps are involved, as in [47]. This conclusion qualitatively agrees with the result found in [36] about the protocol minimizing the heat released in a continuous-time Markov process, when the continuum limit is considered. Consider now a system with a continuous state space, whose dynamics can be described by a continuous ME:…”

Section: Pacs Numberssupporting

confidence: 89%

“…estimating the nonequilibrium activity of biological systems [28] and building efficient molecular engines [29][30][31][32].An enormous amount of studies have investigated how the entropy production gets affected by the coarsegraining [8,[13][14][15]. Some of them have also analyzed the possibility to build a consistent diffusive limit starting from a microscopic picture, containing more information about the system [33][34][35][36]. In [37] it has been shown, in a general setting, that the entropy production estimated using a diffusive description (FPE) is always less (or equal) to the one estimated starting from a more detailed description (ME).…”

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confidence: 99%

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Entropy production in master equations and Fokker–Planck equations: facing the coarse-graining and recovering the information loss

Busiello¹,

Maritan²

2019

J. Stat. Mech.

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Systems operating out of equilibrium exchange energy and matter with the environment, thus producing entropy in their surroundings. Since the entropy production depends on the current flowing throughout the system, its quantification is affected by the level of coarse-graining we adopt. In particular, it has been shown that the description of a system via a Fokker-Planck equation (FPE) lead to an underestimation of the entropy production with respect to the corresponding one in terms of microscopic transition rates. Moreover, such a correction can be derived exactly. Here we review this derivation, generalizing it when different prescriptions to derive the FPE from a Langevin equation are adopted. Then, some open problems about Gaussian transition rates and underdamped limit are discussed. In the second part of the manuscript we present a new approach to dealing with the discrepancy in entropy production due to the coarse graining by introducing enough microscopic variables to correctly estimate the entropy production within the FPE description. We show that any discrete state system can be described by making explicit the contribution of each microscopic current. PACS numbers:A. Entropy Production for coarse-grained dynamics Different levels of description can be adopted to characterize a physical system, depending on the details we are unaware of or deliberately decided to neglect. In particular, when random variables are introduced to encode the uncontrollable degrees of freedom, the energy balance is properly captured by the concepts proper of the stochastic thermodynamics, when the system is small enough [1][2][3]. In this field, a physical system can be described using two different paradigms: Master Equation (ME) and Fokker-Planck equation (FPE) [4][5][6][7]. The former deals with the probability of occupation of each state i at time t, P i (t), and a set of microscopic transition rates between pair of states i and j, W i→j (t). It has the following form:which states the the probability to be in the state i evolves according to the balance between the ingoing and the outgoing probability flux of the state itself. Here we assume that W i→j > 0 implies W j→i > 0, which ensures the ergodicity of the system. Note that here and throughout the manuscript we set W i→i = 0 even though it never enters in the ME above.On the other hand, a Fokker-Planck equation described the system via continuous variables by means of a diffusive dynamics:x (D(x)P (x, t)) (2) where P (x, t) is the probability to be in the state x at the time t, A(x, t) is the drift coefficient and D(x, t) is called the diffusion coefficient. The continuum limit, obtained through a coarse-graining procedure, to derive the latter equation from Eq. (1) is called Kramers-Moyal expansion [4]. It relies on the assumption that the first two "pseudo"-moments [54] of the transition rates are the only ones that are non-negligible, and they are related to A and D respectively.Although the two descriptions appears be equivalent for certain aspects (e.g. equilib...

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Section: Pacs Numberssupporting

confidence: 89%

mentioning

confidence: 99%

Entropy production in master equations and Fokker–Planck equations: facing the coarse-graining and recovering the information loss

Busiello¹,

Maritan²

2019

J. Stat. Mech.

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“…The optimal protocol for an externally controllable gap ǫ(τ ) and given ǫ(0) and ǫ(t) minimizing the mean work W ≡ t 0 dτ p(τ )ǫ(τ ), where p(τ ) is the probability that the energy level is occupied, shows jumps in the optimal ǫ * (τ ) at beginning and end which are nicely explained in physical terms in [264]. A more general approach to the optimal protocol connecting arbitrary given initial and final distributions is given in [265].…”

Section: Underdamped Dynamicsmentioning

confidence: 99%

Stochastic thermodynamics, fluctuation theorems and molecular machines

2012

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Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.

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“…By insisting that the Hamiltonian belongs to the class of admissible controls A, we focused instead on control strategies which we interpret as "macroscopic" in view the regularity assumptions on the control Hamiltonian. These assumptions are analogous to those adopted in previous studies of the entropy production by Langevin-Smoluchowski dynamics [7] or by Markov jump processes [11]. We therefore gather that the existence of the entropy production minimum (54), degenerate because of non-coercivity, and which recovers in the over-damped limit the Monge-Ampère-Kantorovich evolution, yields a robust general picture of the "optimal" thermodynamics for a large class of physical processes described by Markovian evolution equations.…”

Section: Discussionmentioning

confidence: 96%

On extremals of the entropy production by ‘Langevin–Kramers’ dynamics

Muratore-Ginanneschi¹

2014

J. Stat. Mech.

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We refer as "Langevin-Kramers" dynamics to a class of stochastic differential systems exhibiting a degenerate "metriplectic" structure. This means that the drift field can be decomposed into a symplectic and a gradient-like component with respect to a pseudo-metric tensor associated to random fluctuations affecting increments of only a sub-set of the degrees of freedom. Systems in this class are often encountered in applications as elementary models of Hamiltonian dynamics in an heat bath eventually relaxing to a Boltzmann steady state.Entropy production control in Langevin-Kramers models differs from the now wellunderstood case of Langevin-Smoluchowski dynamics for two reasons. First, the definition of entropy production stemming from fluctuation theorems specifies a cost functional which does not act coercively on all degrees of freedom of control protocols. Second, the presence of a symplectic structure imposes a non-local constraint on the class of admissible controls. Using Pontryagin control theory and restricting the attention to additive noise, we show that smooth protocols attaining extremal values of the entropy production appear generically in continuous parametric families as a consequence of a trade-off between smoothness of the admissible protocols and non-coercivity of the cost functional. Uniqueness is, however, always recovered in the over-damped limit as extremal equations reduce at leading order to the Monge-Ampère-Kantorovich optimal mass transport equations.

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Heat Release by Controlled Continuous-Time Markov Jump Processes

Cited by 22 publications

References 54 publications

Entropy production in master equations and Fokker–Planck equations: facing the coarse-graining and recovering the information loss

Entropy production in master equations and Fokker–Planck equations: facing the coarse-graining and recovering the information loss

Stochastic thermodynamics, fluctuation theorems and molecular machines

On extremals of the entropy production by ‘Langevin–Kramers’ dynamics

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