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Discrete State Space Models

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Non-equilibrium Energy Transformation Processes

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Abstract

Consider a two-level system (N = 2) with the general transition rate matrix.

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Notes

  1. 1.

    Here we assume that the transformation exists.

  2. 2.

    Smaller (larger) \(\gamma \) would correspond to saddle points lying closer to level \(k (k + 1)\) in configuration space. The derivations in Sects. 3.4.3 and 3.4.4 can be performed analogously for any \(\gamma \) and \(F_\mathrm{b}\).

  3. 3.

    Force controlled experiments are achievable with magnetic tweezers or with optical tweezers operating in the force clamp mode. In the latter case, the conjugate variable to the applied force \(f\) should better be taken as the trap-pipette distance (rather than the molecular extension) because for this definition the fluctuation theorems are obeyed. Both definitions differ only by a boundary term involving the final and initial forces \(f_\mathrm{f}\) and \(f_\mathrm{i}\), respectively (for details, see [67]).

  4. 4.

    In this case one can show (cf. Sect. 2.3) that \(\langle \exp (-\beta w)\rangle =\phi (t)\exp \{-\beta [F(t)-\mathcal{F}_1(0)]\} =\phi (t)\exp [-\beta F(t)]\), where \(\mathcal{F}_1(0)=0\) from Eq. (3.52) and \(F(t)= -\beta ^{-1}\ln [(A^N-1)/(A-1)]\), \(A=\exp [-\beta (\Delta +vt)]/G\), is the free energy for an equilibrated system with level free energies \(\mathcal{F}_k(t)\) (protocol variables) at time \(t\); \(\phi (t)\) is the probability for the zipper to be in the fully closed microstate \(k=1\) under the reversed protocol, if the system is initially in to the equilibrium state \(\exp \{\beta [F(t)-\mathcal{F}_k(t)]\}\). The situation when the system is initially in thermal equilibrium, and hence the standard Jarzynski equality (2.61) is valid, is discussed in AppendixB. It turns out that in this case, in order to get the tails of the WPD correctly, the refolding can not be neglected.

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Holubec, V. (2014). Discrete State Space Models. In: Non-equilibrium Energy Transformation Processes. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-07091-9_3

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