IMPLEMENTACIÓN DE UN PROGRAMA HACCP EN LA ELABORACIÓN DE CANELONES DE VERDURAS

For the fluctuation symmetry we are interested in the large deviations ofQγ/τ

from its average asτ _{→ ∞}. Such deviations can arise from two sources. First, there are the large deviations of the workWγ, which however, we know, satis-

w* q (c) w w* q (a) w g(w) g(w) w w* q (b) w g(w) w w

Figure 6.12:Three possible scenario’s for the minimum (6.28). In (a) we have the situation

q <−w, so the jump in the derivativeg(w) =I′_{(w) +βθ(q−w) occurs below theg= 0}

axis. Whenq∈[−w, w⋆_{], as can be seen in (b), the jump occurs at theg= 0 axis, so we}

assume the minimum is reached atw =q. For large values ofq, whenq > w⋆_{, the jump}

occurs above theg= 0 axis, so the minimum is reached inw⋆_{, as can be seen in (c).}

also of order τ. This second effect is responsible for the deviations from the standard fluctuation relation (6.2). After all, an energy is typically exponen- tially distributed and we can thus expect a competition with the fluctuations of the work.

In order to clearly see the influence of the unboundedness of the temporal boundary, we consider here the simplest set-up in which deviations from the standard fluctuation symmetry can be calculated exactly.

We consider a particle moving under the influence of a quadratic potential and a random force. For each time stepi= 1,2, . . . , τ, we take the work done on the particleyito be a random variable distributed according to a Gaussian of

average mi and variance vi.8 Let us also consider the analogue of the work

(6.9) as the sum Wτ ≡ (y1+. . .+yτ). By construction, the work per unit

8_{In what follows, we will consider for convenience thatv}

1 = vτ = 1 although a more

time wτ ≡ Wτ/τ is again Gaussian with average wτ = (m1+. . .+mτ)/τ

and varianceσ2

τ = (v1+. . .+vτ)/τ2. If 2wτ =σ2τ, then, automatically, the

probability density function Prob(Wτ =wτ) = Prob(wτ =w) satisfies, for all

τ,

Prob(wτ=w)

Prob(wτ =−w)

=ew,

as was discussed in section 6.3.1 for Gaussian distributions. This is the Gaus- sian analogue of the exact fluctuation symmetry (6.19) for the work, that we here, by the previous construction, assume from the start.

We now consider a new random variable (the analogue of the heat): Qτ(wτ, y1, yτ)≡Wτ+η[(yτ−a)2−(y1−b)2],

wherea, b, ηare real parameters, with density Prob(Qτ =qτ). The aim of our

toy model is to compute

f(q) = lim τ→∞ 1 τ ln Prob(Qτ =qτ) Prob(Qτ=−qτ) .

That can follow fromf(q) =h(q)₋h(₋q) with h(q) the large deviation rate function of the heat: Prob(Qτ =qτ)≃exp(τ h(q)). The large deviation rate

functionh(q) is also the Legendre transform of the generating function E(t) = lim τ→∞ 1 τlnEτ(t) with Eτ(t) =E e tQτ₌ 1 (2π)32det 1 2C Z dy etQτ(y)_e−21(y−y¯)·C −1_(y−y¯) , (6.29) where, collectively,y = (wτ, y1, yτ) and ¯y≡(wτ,y¯1,y¯τ) represent their mean

while C = Cτ corresponds to the covariance matrix of y. Computing the

Gaussian integrals in (6.29) and taking the limitτ→ ∞leads to

E(t) =    1 2vt 2_{+tw, ift} ∈[₋t⋆, t⋆], +_∞, otherwise, wheret⋆= 1/2η andv= limτσ2τ = 2w= 2 limτwτ.

We are now interested in evaluating the Legendre transform of the above, h(q) =₋supt[qt−E(t)]. The location of the supremum depends on whether

(q₋w)/vlies within or outside the interval [₋t⋆, t⋆]. As a result,h(q) becomes

a quadratic function within the interval [₋vt⋆+w, vt⋆+w] and a linear one

two cases depending on the value ofw. Forvt⋆< w, one has f(q) =          2qt⋆, for q∈[0, w−vt⋆], −1 2v(q−w)2+qt⋆− 1 2vt2⋆+wt⋆, for q∈[w−vt⋆, w+vt⋆], 2wt⋆, for q>w+vt⋆.

while forw < vt⋆, one has

f(q) =          q, for q_∈[0,₋w+vt⋆], −1 2v(q−w)2+qt⋆− 1 2vt2⋆+wt⋆, for q∈[−w+vt⋆, w+vt⋆], 2wt⋆, for q>w+vt⋆. (6.30) The results of section 6.3.2 and of reference [122], i.e., the Gaussian case where β = 1, are reproduced in the case (6.30) by choosingw= 1 andt⋆ = 1, i.e.,

η= 1/2.

6.7

The basis of a Jarzynski relation

For completeness, we review a discussion by Maes [79] on the Jarzynski equal- ity, which connects the discussion of time-reversal and entropy production of section 2.3.2 with the fluctuation theorems and related subjects as discussed in this chapter.

Let Γ be the phase space on which we have a time-dependent dynamics defined in terms of invertible transformationsft. One can think of a protocol γ that

changes in discrete steps so that a phase space pointx_∈Γ flows in timet to ϕt,γx∈Γ with

ϕt,γ =ft. . . f2f1, t= 1, . . . , τ.

For the reversed protocol Θγ, this yields

ϕt,Θγ =fτ−t+1 . . . fτ−1fτ.

We imagine a measure µ on the phase space Γ that is left invariant byϕt,γ:

µ(ϕ−t,γ1B) =µ(B) for B ⊂Γ. Furthermore, Γ is equipped with an involution

πthat also leavesµinvariant. We assume dynamical reversibility in the sense that for allt,

ftπ=π ft−1,

as was also mentioned in equation (1.5). As a consequence, we have that π ϕ−_t,Θ1_γπ=fτ . . . fτ−t+1or ϕ−τ,γ1π ϕ−t,Θ1γπ=ϕ

−1 τ−t,γ.

Let us now divide the phase space in a finite partition ˆΓ. It corresponds to a reduced description; each element in the partition is thought to reflect some manifest condition of the system. The entropy is defined `a la Boltzmann as

For example, in Hamiltonian systems one takes the Liouville measure as the invariant measureµ, and then we obtain the conventional Boltzmann definition S = ln|M|. We fix probability laws ˆρand ˆσ on the elements of the partition and we specify the initial probability measure on Γ as

rρˆ(A)≡

X

M

µ(A_∩M) µ(M) ρ(Mˆ ).

This probability measuresA⊂Γ using ˆρat the level of the partitionsM of the reduced description and using the invariant measureµwithin each partitionM. The reduced trajectories of the system are sequences ω= (M0, M1, , . . . , Mτ)

where Mi ∈ Γ, indicating subsequent moments when the phase space pointˆ

was in the set Mi, i= 0, . . . , τ. The path-space measure Prob(ω|γ)ρˆgives the

probability of trajectories when starting fromr(ˆρ) and using protocolγ. The quantity of interest that measures the irreversibility in the dynamics on the level of ˆΓ is (see also the Crooks relation (6.23) and section 2.3.2):

R= ln Prob(M0, M1, . . . , Mτ|γ)ρˆ Prob(πMτ, πMτ−1, . . . , πM0|Θγ)σπˆ

.

The point is that for every probability ˆρand ˆσon ˆΓ, and for allM0, . . . , Mτ∈

ˆ Γ,

R=S(Mτ)−S(M0)−ln ˆσ(Mτ) + ln ˆρ(M0)

To prove this, we only have to look closer at the consequences of the dynamic reversibility. By using thatµ(B) =µ(ϕ−τ,γ1πB), we have of course that

µ " _τ \ t=0 ϕ−t,Θ1γπMτ−t # =µ " _τ \ t=0 ϕ−τ,γ1π ϕ−t,Θ1γπMτ−t # , but moreover, by reversibility, the last expression equals

µ " _τ \ t=0 ϕ−τ,γ1π◦ϕ−t,Θ1γπMτ−t # =µ " _τ \ t=0 ϕ−τ−1t,γMτ−t # , which is all that is needed.

As an immediate corollary, under the expectation Prob(ω|γ)ρˆ, one has

e−S(Mτ)+S(M0)+ln ˆσ(Mτ)−ln ˆρ(M0)_{= 1. (6.31)}

A simple choice for the system and partition takes an isolated system where the reduced variablesMi refer to the energy of the system. We still have the

freedom to choose ˆρ and ˆσ. Let us take ˆρ(M0) = 1 where indeed M0 refers

distributed on the energy shellE′_{. For these choices, in ‘suggestive’ notation,} equation (6.31) becomes ln ProbE(E→E ′_|γ) ProbE′(E′ →E|Θγ) =S(E′)₋S(E) Using that here ∆E=E′

−E =W equals the work done, one thus recovers the microcanonical analogue of the Crooks relation (6.23), see also [23]. The mathematically trivial identity (6.31) is the mother of all Jarzynski rela- tions. The way in which it gets realized as for example an irreversible work-free energy relation depends on the specific context or example. We can also split the system from the environment. The reduced variablesMi can for example

be chosen to consist of the microscopic trajectory for the system and of the se- quence of energies of the environment. For a thermal environment at all times in equilibrium at inverse temperature β, we thus getS(Mτ)−S(M0) = βQ

whereQis the heat that flowed into the reservoir. On the other hand, we can take ˆρand ˆσas equilibrium distributions, say of the weak coupling form

ˆ

ρ(M) = e

−βU(x,γ0)

Z0

h(E),

where M = (x, E) combines the micro-statexof the system and the energy E of the environment,h(E) describes the reservoir-distribution,U(x, γ) is the energy of the system with parameterγ. Similarly,

ˆ

σ(M) = e

−βU(x,γτ) Zτ

h(E).

If we have thath(E0)≃h(Eτ), i.e., that the energy exchanges to the environ-

ment remain small compared to the dispersion of the energy distribution in the reservoir, we get from equation (6.31) in that context that

he−βQ−βU(xτ,γτ)+βU(x0,γ0)

iρˆ=

Zτ

Z0

, which is a version of the Jarzynski relation (6.25).

6.8

Conclusions

In this chapter, we studied how the heat and (dissipative) work behave when a particle is dragged in a potential Ut(x), following a protocolγ. This study

is relevant to understand how biological and nanoscale devices are able to perform useful work and how much heat is dissipated.

• For the work, a general framework was presented that unites the studies of several authors, who concentrated on particular potentials with par- ticular protocols through explicit calculations. We formulated a sharp

condition (6.18) on the potential and protocol for which the existence of an exact fluctuation theorem for the work is found. The numerical work from section 6.4 suggests that any deviation from this condition leads to a violation of the EFT. The calculations from section 6.5.2 emphasize the importance of the breaking of time-reversal invariance to understand how the symmetries in the probability density of the (dissipative) work arise.

• When working with bounded potentials9_{, it is clear that there is no prob-}

lem in proving a fluctuation theorem for the heat for long observation times τ _{→ ∞}. After all, the heat production per unit time converges in probability to the work done per unit time, and we know the latter satisfies the EFT.

As expected by the work of van Zon and Cohen [121, 123], this is no longer necessarily true for unbounded potentials. Corrections to the fluctuation theorem are possible when the particle is “egocentric” and stores the work that is being done on it as potential energy instead of dissipating it as heat. In this case, the particle explores the boundaries of the potential and prevents ∆U/τ of going to zero.

We argued in this chapter that the results of van Zon and Cohen are also expected beyond the harmonic potential that they studied. As they hypothesized, the key to obtaining corrections to the FT is that for large values ofQ=W ₋∆U, the exponential tails of Prob(∆U =u)_∼e−τ|u| give a larger contribution than the smooth tails of the work distribution, and hence Prob(Q = q) behaves exponential for large q. The correc- tion obtained in this way is generic: for small fluctuations, the regular fluctuation theorem holds, i.e., Prob[Q = qτ] ≈ eβqτ_{Prob[Q = −qτ].}

For large fluctuations, the exponential damping vanishes and we get Prob[Q=qτ]≈e2βτ_{Prob[Q=−qτ].}

However, the major conclusion is that the global scheme of looking at the time- reversal breaking [78, 79, 80] works also in this context and gives useful infor- mation about dissipative physical quantities such as heat, dissipative work and entropy production: we now know precisely how the typicality of irreversibil- ity is attained. In that sense, the study of fluctuation theorems contributes to the understanding of irreversibility and the construction of nonequilibrium statistical mechanics.

9_{On spatially finite domains: bounded potentials on}

Conclusions and outlook

In this thesis, we have discussed some of the emergent properties of large, but finite, classical and quantum systems, and in what sense these properties can be understood in smaller systems which are more vulnerable to fluctuations. The general framework used is that of studying the time-reversal symmetry at the level of trajectories, as first introduced in the papers [80, 81]. In a very straightforward way, this framework yields results regarding

• the manifestation of entropy as a measure of time-reversal symmetry breaking and of macroscopic irreversibility,

• symmetry properties for the probability distributions of heat and work. The relation between entropy and time-reversal symmetry breaking was al- ready known for situations involving classical mechanics, and with this thesis, a step towards a similar quantum mechanical set-up was attempted. The ex- tended fluctuation symmetries for work and heat had been previously estab- lished for the harmonic case, and with this thesis, the case of a more general potential and protocol was derived. We briefly summarize our conclusions and point out where extensions and future work are possible.

7.1

Reversibility and irreversibility in the quan-

tum formalism

In chapter 3, we have extended the existing results for classical systems to a simple quantum set-up regarding the relation between time-reversal and en- tropy production.

The first result concerns a free quantum evolution, essentially prescribed by the Schr¨odinger equation, that starts from a macroscopically prepared distribution ˆ

µ(α) on the macrostatesα. After a timet, we perform a (set of) macroscopic measurement(s) to determine the final macroscopic distribution ˆµt(α). We

certain macrostate αt at the end of the experiment, when started from α0.

In equation (3.5), we found that the fraction of the time-forward probability Probµˆ[α0, αt] to the time-reversed probability Probˆµtπ[παt, πα0] is given by

ln Probµˆ[α0, αt] Probˆµtπ[παt, πα0]

=SB(αt)−SB(α0)−ln ˆµt(αt) + ln ˆµ(α0),

where SB(α) is the (quantum) Boltzmann entropy (2.20). When taking the

expectation value with respect to the initial distribution ˆµ, we recover the change in Gibbs-von Neumann entropies (2.27):

Eµˆ ln Probµˆ[α0, αt] Probµˆtπ[παt, πα0] =SG(ˆµt)−SG(ˆµ).

This result can be generalized to a free quantum evolution which is interrupted by measurements. In this way, we get probability distributions Probµˆ[ω] at the

level of a trajectoryω= (α0, α1, . . . , αt) of macrostatesαj. Then, the ratio of

a macroscopic trajectory ω to the time-reversed trajectory Θω is almost the difference of Boltzmann entropies

ln Probµˆ[ω] Probµˆtπ[Θω]

=SB(αt)−SB(α0)−ln ˆµt(αt) + ln ˆµ(α0).

The expectation value with respect to Probµˆ[ω] gives

Eµˆ ln Probµˆ[ω] Probµˆtπ[Θω] =SG(ˆµt)−SG(ˆµ),

which is again the difference in Gibbs-von Neumann entropies of the initial and final macroscopic distributions.

The two results of above show that at the level of probabilities for trajecto- ries, the inclusion of the quantum measurement in the dynamics does not give rise to a fundamental difference. In the standard statistical interpretation of quantum mechanics, we get a manifestation of statistical reversibility, quite similar to the situation of detailed balance for transition probabilities. This statistical reversibility is also found in classical Hamiltonian mechanics, and as the results of above show, also satisfied within the standard theory of quan- tum mechanics. Consequently, the quantum measurement, even though it is fundamentally irreversible, does not not need to be invoked for macroscopic irreversibility. This was argued in chapter 4.

In chapter 5, we used a simple quantum model to argue that, as in the classical case, thermodynamic irreversibility is a statement about thetypical temporal behaviour of macroscopic observables. We considered a system of N distin- guishable spin 1/2 particles, arranged on a ring and rotating against a fixed set of scatterers. As the number of particles of the system gets very large, and when the system is observed over the appropriate time scales, the evolution

of the magnetization as a function of time, typically follows an autonomous equation (5.31): ~ mt≡m~e+ Re " ~ m0−m~e+ i(~h_×m~0) h 1₋µ+µexp[i2h]t # , wherem~e is the equilibrium magnetization,m~0 the initial magnetization,µis

the fraction of scatterers on the ring andH =~h·~σdefines the single-site Hamil- tonian as written in function of the Pauli matrices. This autonomous evolution at the level of macrostates, gives rise to an H-theorem 5.4.2, i.e., the Gibbs- von Neumann entropy increases as a function of time. This is an example of the quantum mechanical generalization of the Jaynes arguments of section 2.4.3, which predict the existence of an H-theorem in the classical Hamilto- nian case. It should be stressed that it was important for theH-theorem to look at the proper macroscopic variables, which are the three magnetization components in this case. When restricting the description to only one com- ponent of the magnetization vector, for example the z-component, one can get an autonomous equation for some special initial conditions. However, the corresponding Gibbs-von Neumann entropy does not satisfy anH-theorem in that case.

All this shows that the emergence of irreversibility in the quantum case has the same origins as in the classical case: it is an interplay of scale separation between the micro/macroworld, appropriate timescales and suitable initial con- ditions.

7.2

Fluctuation relations and irreversibility

The study of so-called fluctuation relations for the heat and work, was first initiated by the work of van Zon and Cohen [122], who considered a Brownian particle which is dragged through a fluid by a uniformly moving harmonic op- tical trap. Through explicit calculation, they obtained a fluctuation theorem for the work and a correction to the fluctuation theorem for the heat. Their work was a direct motivation for the results presented in chapter 6 of this thesis. Our first result translates the fluctuation theorem for the work into the lan- guage of time-reversal symmetry breaking, and at the same time it relates to this symmetry breaking two well-known nonequilibrium relations: the Crooks

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