Recovery of Free Energy Branches in Single Molecule Experiments

Junier, Ivan; Mossa, Alessandro; Manosas, Maria; Ritort, Fèlix

doi:10.1103/physrevlett.102.070602

Cited by 68 publications

(100 citation statements)

References 12 publications

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“…We can thus define a conditional equilibrium free energy F leq (t), which is the free energy of the system given that it is in the left well [43,44]. In analogy with the usual definition of the equilibrium free energy, we have,…”

Section: Theory Second Law Of Thermodynamics In Terms Of Workmentioning

confidence: 99%

“…That is the system is in state L (left) with probability p(t) and state R (right) with probability 1 − p(t). But, constrained to be within one well or the other, the system is in thermal equilibrium.We can thus define a conditional equilibrium free energy F leq (t), which is the free energy of the system given that it is in the left well [43,44]. In analogy with the usual definition of the equilibrium free energy, we have,…”

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confidence: 99%

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Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Gavrilov

Chétrite

Bechhoefer

2017

Proc. Natl. Acad. Sci. U.S.A.

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Stochastic thermodynamics extends classical thermodynamics to small systems in contact with one or more heat baths. It can account for the effects of thermal fluctuations and describe systems far from thermodynamic equilibrium. A basic assumption is that the expression for Shannon entropy is the appropriate description for the entropy of a nonequilibrium system in such a setting. Here, for the first time, we measure experimentally this function. Our system is a micron-scale colloidal particle in water, in a virtual double-well potential created by a feedback trap. We measure the work to erase a fraction of a bit of information and show that it is bounded by the Shannon entropy for a twostate system. Further, by measuring directly the reversibility of slow protocols, we can distinguish unambiguously between protocols that can and cannot reach the expected thermodynamic bounds. INTRODUCTIONBeginning with the foundational work of Clausius, Maxwell, and Boltzmann in the 19th c., the concept of entropy has played a key role in thermodynamics. Yet, despite its importance, entropy is an elusive concept [1][2][3][4][5][6][7][8], with no unique definition; rather, the appropriate definition of entropy depends on the scale, relevant thermodynamic variables, and nature of the system, with ongoing debate existing over the proper definition even for equilibrium cases [9]. Moreover, entropy has not been directly measured but is rather inferred from other quantities, such as the integral of the specific heat divided by temperature. Here, by measuring the work required to erase a fraction of a bit of information, we isolate directly the change in entropy in an open nonequilibrium system, showing that it is compatible with the functional form proposed by Gibbs and Shannon, giving it a physical meaning in this context. Knowing the relevant form of entropy is crucial for efforts to extend thermodynamics to systems out of equilibrium.For a continuous classical system whose state in phase space x is distributed as the probability density function ρ(x), the Gibbs-Shannon entropy is [10,11] where k B is Boltzmann's constant. For quantum systems, von Neumann introduced, in 1927, the corresponding expression in terms of the density matrix [12]. Historically, the system in Eq. 1 has typically been assumed to be in thermal equilibrium. * email: johnb@sfu.caThe physical relevance of Eq. 1 for a nonequilibrium distribution ρ(x) has often been questioned (e.g., [13][14][15][16][17]). One concern is that S is constant on an isolated Hamiltonian system and can change only when evaluated on subsystems, such as those picked out by coarse graining. With many ways to choose subsystems or to coarse grain, is the associated notion of irreversibility intrinsic to the description of the system?In another approach to entropy, advanced in the context of communication and information theory, Shannon [11,18] proved that, up to a multiplicative constant, S is the only possible function satisfying three intuitive axioms. Alternatively, one can start from an...

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Section: Theory Second Law Of Thermodynamics In Terms Of Workmentioning

confidence: 99%

mentioning

confidence: 99%

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Gavrilov

Chétrite

Bechhoefer

2017

Proc. Natl. Acad. Sci. U.S.A.

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“…The existence of a maximal deviation from equilibrium for which the reconstruction of ∆F is possible is well known from previous studies [20][21][22][23]. In the context of unfolding of a large molecule this was typically viewed as an event of a "single particle" escaping from one well into another [35], or a sequence of such events [13]. We attempted to integrate the well-known static and dynamic scaling properties of polymers into the description of their non-equilibrium motion.…”

Section: Discussionmentioning

confidence: 99%

Nonequilibrium interactions between ideal polymers and a repulsive surface

Levi

Kantor

2017

Phys. Rev. E

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We use Newtonian and overdamped Langevin dynamics to study long flexible polymers dragged by an external force at a constant velocity v. The work W performed by that force depends on the initial state of the polymer and the details of the process. The Jarzynski equality can be used to relate the non-equilibrium work distribution P (W ) obtained from repeated experiments to the equilibrium free energy difference ∆F between the initial and final states. We use the power law dependence of the geometrical and dynamical characteristics of the polymer on the number of monomers N to suggest the existence of a critical velocity vc(N ), such that for v < vc the reconstruction of ∆F is an easy task, while for v significantly exceeding vc it becomes practically impossible. We demonstrate the existence of such vc analytically for an ideal polymer in free space and numerically for a polymer which is being dragged away from a repulsive wall. Our results suggest that the distribution of the dissipated work W d = W − ∆F in properly scaled variables approaches a limiting shape for large N .

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“…Since partially equilibrated states are often observed in several complex systems such as proteins, the present scheme might be useful for experimental determination of the free energy of such states [33].…”

Section: Under-damped Casesmentioning

confidence: 99%

Work relations for time-dependent states

Nakagawa

Sasa

2013

Phys. Rev. E

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For time-dependent states generated by an external operation, a generalized free energy may be introduced by the relative entropy with respect to an equilibrium state realized after sufficient relaxation from the time-dependent states. Recently, by studying over-damped systems, Sivak and Crooks presented a formula that relates the generalized free energy with measurable thermodynamic works. We re-derive this relation with emphasizing a connection to an extended Clausius relation proposed in the framework of steady state thermodynamics. As a natural consequence, we generalize this relation to be valid for systems with momentum degrees of freedom, where the Shannon entropy in the generalized free energy is replaced by a symmetric one.

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Recovery of Free Energy Branches in Single Molecule Experiments

Cited by 68 publications

References 12 publications

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Nonequilibrium interactions between ideal polymers and a repulsive surface

Work relations for time-dependent states

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