The Heat Distribution of the Underdamped Langevin Equation
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b The Heat Distribution of the UnderdampedLangevin Equation.
Pedro V. Paraguass´u
Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica22452-970, Rio de Janeiro, Brazil
Rui Aquino
Departamento de F´ısica Te´orica, Universidade do Estado do Rio de Janeiro,20550-013, Rio de Janeiro, Brazil
Welles A. M. Morgado
Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica22452-970, Rio de Janeiro, Braziland National Institute of Science and Technology for Complex Systems, BrazilE-mail: [email protected]
Abstract.
In the Stochastic Thermodynamics theory, heat is a random variable witha probability distribution associated. Studies in the distribution of heat are mostlyin the overdamped regime. Here we solve the heat distribution in the underdampedregime for three different cases: the free particle, the linear potential, and the harmonicpotential. The results are exact and generalize known results in the literature.
1. Introduction
Stochastic Thermodynamics [1, 2, 3, 4] is a recently developed field that connectsthermodynamics with the fluctuation world of small out-of-equilibrium systems. Overthe past two decades, a lot of results were made available due the discovery of fluctuationtheorems [5, 6, 7, 8, 9], which are constraints in the non-equilibrium distribution ofthermodynamics quantities like work, heat, and entropy.Besides the general discoveries of fluctuation theorems, characterizing a systemin a non-equilibrium regime needs more effort than just finding constraints for theprobabilities. Heat and other thermodynamic functionals are random variables, and thenhave associated probability distributions. The calculation of the probability distributionof a thermodynamic functional is in itself a difficult task, and only a few analytical resultsare possible. Exact results are found in the overdamped regime [10, 11, 12, 13, 14, 15],where the inertia can be neglected due the strong friction. For the underdamped limit he Heat Distribution of the Underdamped Langevin Equation. he Heat Distribution of the Underdamped Langevin Equation.
2. Stochastic Thermodynamics of Brownian Motion
The Brownian particle is one of the most standard systems in the domain of StochasticThermodynamics. Before deriving the heat distribution for 3 different cases, we give abrief review of the thermodynamics of such system.The Brownian particle, which are in contact with a heat bath, evolves in timeaccording to the generalized Langevin Equation [35] m ¨ x = − γ ˙ x + φ ( x, t ) + η ( t ) . (1)This corresponds to Newton ´ s second law of classical mechanics with a stochastic nature,due to the bath force η ( t ). This force models the interaction between the particle andthe fast degrees of freedom of the bath [36]. For a thermal bath, the paradigmatic caseof Stochastic Thermodynamics, this force is modeled by a Gaussian white noise. Then,the bath generated force has the statistical properties (all other cumulants being null): h η ( t ) η ( t ′ ) i = 2 γT δ ( t − t ′ ) , h η ( t ) i = 0 , (2)where T is the temperature of the heat bath, and γ is the frictional coupling constant.Note that we are working here with k B = 1. The other two forces on the right side in(1) are the drift force − γ ˙ x which comes from the slow degrees of freedom of the bath,and the force φ ( x, t ) which can account for generic external forces, and also for potentialforces applied to the particle, i.e., φ ( x, t ) = F ( t ) − ∂V∂x ( x, t ) , (3)where, F ( t ) is an external non-conservative force acting on the particle, and − V ′ ( x, t )is a force derived from an internal potential V ( x, t ). The time dependence in theconservative potential allows for the inclusion of an external protocol in the parametersof the potential. As an example, F ( t ) can be an external electric field generated by acapacitor-like device, with time variation of the voltage [29, 37], and − V ′ ( x, t ) can bea harmonic potential where the spring constant might vary in time according to someprotocol [25, 26]. These are similar examples of the systems that will be studied in thenext section.For the Stochastic Thermodynamics of the above system, we start defining the heatexchanged between the particle and the bath. Following Sekimoto[2] we have Q [ x ( τ )] = Z τ ( − γ ˙ x + η ( t )) ˙ xdt, (4)where the first term takes into account the energy lost by the particle to the reservoir,while the second term is the energy acquired by the particle from the same reservoir. he Heat Distribution of the Underdamped Langevin Equation. x ( t ) gives a stochastic nature tothe heat. The stochastic nature of the thermodynamic variables, such as heat, work,and entropy, is what explains the name Stochastic Thermodynamics [3]. The aboveformula represents the heat exchange during a time interval of t ∈ [0 , τ ]. Using theLangevin Eq. (1) we have Q [ x ( τ )] = Z τ ( m ¨ x − F ( t ) + V ′ ( x, t )) ˙ xdt = 12 m ( v t − v ) + Z τ ( V ′ ( x, t ) ˙ x − F ( t ) ˙ x ) dt. (5)Notice that Q [ x ( τ )] depends on the variation of the kinetic energy, where we define˙ x ( τ ) = v ( τ ) = v t , ˙ x (0) = v . Herein, we only deal with cases where the conservativepotential are time independent V ( x, t ) → V ( x ). In this case, the heat becomes Q [ x ( τ )] = 12 m ( v t − v )+∆ V + Z τ F ( t ) ˙ xdt = ∆ E + W [ x ( t )] (6)where in the second equality we define,∆ E = 12 m ( v t − v )+∆ V + F ( τ ) x τ − F (0) x , as the variation of the internal energy of the particle, and [7] W [ x ( t )] = Z τ x dF, (7)as the external work done on the particle due the external force. Note that (6) is thefirst law of thermodynamics.Herein, we shall concentrate in characterizing the heat in cases where we haveonly the contribution from the internal energy, i.e., F = 0. (we only have theforce corresponding to φ = − ∂ x V , and V ( x, t ) = V ( x )) The fact that the functionaldependence of Q [ x ( τ )], comes just from the internal energy, gives us the idea that wewill have a trivial result. However, the boundary terms are random variables as wells,giving a non-trivial result for the distribution.
3. Path Integral Formalism
In order to derive the heat distribution we need to solve a path integral. We give, in thissection, the steps of the path integral formalism of stochastic process [20, 21, 19, 24],with the simplifications for the cases studied in this article.The transitional probability between x , v and x t , v t , of the Brownian motiondescribed by (1) is given by the path integral [38, 24] P [ v t , x t , τ | v , x ,
0] = Z τ Dx exp − γT A [ v, x ] ! . (8)The stochastic thermodynamics is formulated in the Stratonovich prescription [39]. Formost of the cases the stochastic action will have the structure A [ v, t ] = Z τ η [ v, t ] dt (9)where now, the bath force η can be seen as a ”functional of the position and velocity”,since, from (1), η → η [ x ( t )] = m ¨ x + γ ˙ x − φ ( x, t ). In the Stratonovich prescription he Heat Distribution of the Underdamped Langevin Equation. x ( t ) = x c ( t ) + y ( t ), where x c ( t ) have the sameboundary conditions of x ( t ), and is the solution of the extremization δ A = 0. Makingthis transformation in the action, we find P [ v t , x t , τ | v , x ,
0] = exp − γT A [ v c , x c ] ! Z Dy e A [ y ] . (10)Instead of solving the path integral in y , we can just use the normalization property ofthe transitional probability, that is Z dv t dx t P [ v t , x t , τ | v , x ,
0] = 1 (11)This gives the constraint Z Dy e A y = Z dv t dx t exp − γT A [ v c , x c ] !! − . (12)This happens because y ( t ) does not depend on the boundary constants v t , x t , v , x . It’simportant to notice that this simplification only occurs because we have a quadraticaction. For more general systems this method only gives us an approximation, and thepath integral in y ( t ) needs to be solved [40]. The transitional probability will be now P [ v t , x t , τ | v , x ,
0] = exp (cid:16) − γT A [ v c , x c ] (cid:17)R dv t dx t exp (cid:16) − γT A [ v c , x c ] (cid:17) (13)In the Appendix A and Appendix B we solve the path integral for the case 2 and 3.
4. The Heat Distribution
Heat is a stochastic variable as much as the position of the particle. To derive theprobability distribution for the heat, we shall use path integral techniques. We shall seethat the derived distribution is not trivial and, in two of the three cases, we need to usenumerical integration.
Consider the equation (1) without any force unless the two forces that comes from thebath, that is m ˙ v + γv = η ( t ) , (14) he Heat Distribution of the Underdamped Langevin Equation. v = ˙ x is the velocity of the particle. Note that we can just write the Langevinequation in terms of the velocity, giving a more simple stochastic differential equation.The equation 14 is the original equation found by Paul Langevin in 1908 [27].Despite the time, one can ask if remains new physical insights to be discovered usingthis equation. Here, using the stochastic thermodynamics framework, we will show thestatistical distribution for the heat associated with Eq. 14.The heat for (14) is just the variation of the kinetic energy Q [ v ( τ )] = 12 m ( v t − v ) . (15)The Langevin Eq. 14 and the Eq. 15 for the heat, are mathematically equivalent to theOrnstein-Uhlenbeck equation [42] and heat of a harmonic potential in the overdampedregime. Such overdamped case was solved by Charttejee and Cherayil [10], and thesteps used here are essentially the same.Due to the functional dependence on the stochastic variable v ( t ), heat is alsoa stochastic variable and has an associated distribution. This distribution can becalculated by [19, 17] P ( Q, τ ) = h δ ( Q − Q [ v ( τ )]) i = Z dλ π e iλQ Z dv t Z dv P ( v ) Z v ,v t Dv e A [ v ( t )] − iλQ [ v ( τ )] , (16)where the average are in all trajectories in the interval t ∈ [0 , τ ]. The exponent A [ v ( t )]is the “Stochastic Action” or Onsager-Machlup Lagragian [38], defined in this case by A [ v ( t )] = − γT Z ( m ˙ v + γv ) dt. (17)The path integral in Eq. 16 is the conditional probability for the velocity, that is Z v ,v t Dv e A [ v ( t )] = P [ v t , t | v , , (18)which is a Gaussian path integral, with known result [24, 19, 41]: P [ v t , τ | v ,
0] = vuut m (cid:16) coth (cid:16) γτm (cid:17) + 1 (cid:17) πT exp − m (cid:16) v − e γτ/m v t (cid:17) ( − γτ /m ))4 T (19)Since in Eq. 16 the exponent of iλQ [ v ( τ )] only depends on the boundary points, it canbe removed from the path integral. We are left with Gaussian integrals in v and v t ,with P ( v ) as the Boltzmann distribution. That is P = m / (2 πT ) − / exp( − mv / T ).The integral in v t and v is solved, giving Z dv t Z dv P ( v ) P [ v t , t | v , e − iλQ [ v ( τ )] = r(cid:16) coth (cid:16) γτm (cid:17) + 1 (cid:17)r coth (cid:16) γτm (cid:17) + 2 λ T + 1 (20)We are left with the integral in λ , which can be solved using the integral representationof the modified Bessel function of the second kind of zero order [43] K ( z ) = Z ∞−∞ cos ( zt )( t + 1) − / . he Heat Distribution of the Underdamped Langevin Equation. Figure 1.
Heat Distribution for the free particle. The parameters chosen are τ = m = γ = T = 1. This gives the heat distribution, P ( Q, τ ) = s(cid:18) coth (cid:18) γτm (cid:19) + 1 (cid:19) Z ∞−∞ dλ π e iλQ r coth (cid:16) γτm (cid:17) + 2 λ T + 1 == r coth (cid:16) γτm (cid:17) + 1 π √ T K | Q | r coth (cid:16) γτm (cid:17) + 1 √ T . (21)The result (21) is the heat distribution for an underdamped free particle in contactwith a heat bath of temperature T , and is plotted in figure 1. This distribution ismathematically equivalent of the heat distribution of an overdamped particle diffusingin a harmonic potential [10]. Due the modified Bessel function, we have an exponentialtail in the distribution, as expected [33].With (21), we can try recover the overdamped limit. For m/γ << m → P ( Q, t ) = 1 πT K | Q | T ! . (22)However, this is not the expected result. For the free particle, if we take the overdampedlimit in the dynamics (which corresponds to taking simply m = 0, which does notgenerally lead to the same result as taking m/γ → γv = η , giving for the heat Q [ v ] = 0. It is easy to check that the expecteddistribution in this case will be P ( Q ) = δ ( Q ). There is then a difference betweentaking the overdamped limit in the dynamics first, than taking the limit in the heat he Heat Distribution of the Underdamped Langevin Equation. τ → ∞ , we will find this same result. The simplest interaction that we can perform on the particle is an external force F ( t ) = κ . In this case the Langevin Equation becomes m ¨ x + γ ˙ x = η ( t ) + κ. (23)There are examples of this type of force in the literature [45], a very simple one being thecharged Brownian particle inside a capacitor with uniform electric field [29]. Followingagain the definition of Sekimoto [2], the heat functional will be Q [ x ( τ )] = 12 m ( v t − v ) − κ ( x t − x ) . (24)Note that, since a dependence in x ( t ) appears in Eqs. (23) and (24), we cannot ignore theposition probability in the calculus of the heat distribution. Now we need to integrateboth on the position and the velocity.The heat distribution will be given by P ( Q, τ ) = Z dλ π e iλ ( Q − m ( v t − v )+ κ ( x t − x ) ) Z d Γ t Z d Γ P (Γ ) Z v ,v t ,x t ,x Dx e A [ x ( t )] , (25)where Γ (0 ,t ) = { x (0 ,t ) , v (0 ,t ) } are the boundary terms. Now the Stochastic Action will be A [ x ( t )] = − γT Z τ ( m ¨ x + γ ˙ x − κ ) dt. (26)The path integral for this action is solved following Section 3, and the steps are given indetail in the Appendix A. The solution of the path integral gives a quadratic dependencein the boundary terms, that is Z Dx exp − γT Z τ ( m ¨ x + γ ˙ x − κ ) dt ! = N exp ( f ( v t , v , x t , x )) (27)Despite the second derivative inside the action, the path integral in the above equationis already known in the literature in the context of quantum mechanics [23]. Thenormalization constant N and the function f are given in the Appendix A. Thedependence of the boundary conditions is written in the term f , and is a quadraticpolynomial in the boundary terms. This yields a Gaussian integral over x , x t , v , v t .Now we are left only with the integrals over the boundary conditions and theparameter λ in (25). The integrals over the boundary terms are Gaussian and can becomputed. First the integral in x t gives (ignoring the multiplicative terms) Z d Γ dv t Z dx t exp (cid:20) f (Γ t , Γ ) + iλ (cid:18) κ ( x t − x ) − m v t − v ) (cid:19)(cid:21) ∼ (28) ∼ exp tanh (cid:16) γτ m (cid:17) (cid:16) γκτ (1 + 2 iλT ) coth (cid:16) γτ m (cid:17) + κm ( − − iλT ) + γm ( v + v t ) (cid:17) γ T (cid:16) γτ coth (cid:16) γτ m (cid:17) − m (cid:17) he Heat Distribution of the Underdamped Langevin Equation. x t , we notice that there is no dependence in x . Thismeans that we can neglect the integration in x , just using that the probability P ( x , v )satisfies R P ( x , v ) dx = P ( v ). Assuming an initially thermalized distribution for thevelocity, we have P ( v ) = r m πT exp (cid:18) − T mv (cid:19) . (29)The subsequent integration in v t , v will be straightforward, since it is just a Gaussianintegral. Integrating in v t and v , and collecting the multiplicative terms, gives thecorrect result for the heat distribution: P ( Q, τ ) = e γτm π I ( Q, τ ) = e γτm π Z dλ exp ( iλQ + f ( λ )) r cosh (cid:16) γτm (cid:17) + (2 λ T + 1) sinh (cid:16) γτm (cid:17) (30)where the function f ( λ ) is given in the Appendix A. The dependence on λ for the f isnot trivial: the integral in (30) is similar to (21), where it yields a Bessel function as theresult. However, the λ -dependence in f generates a complicated function, which doesnot satisfy the Bessel definition. Therefore, the integral in (30) needs to be integratenumerically. Using Wolfram Mathematica [46] we were able to integrate it numerically,for any values of the parameters of the system. The plot of the distribution P ( Q, τ ) atthree different times is given in figure 2.It can be shown that in the limit κ = 0 the distribution in (30) converges to thedistribution of case 1, (21). The function f vanishes in that limit, while the denominatorcan be rewritten to match with (21). This is a consistency test for the found distribution,since it is a simple generalization of the former case. Let’s consider now the case where we have a harmonic potential in (1), that is, V ( x, t ) = kx /
2. This is an important case, since most of the experiment’s with opticaltweezers can be modeled by a harmonic potential [45, 25, 18]. As an example, forthe Brownian Carnot engine [26], one has to consider this potential combined with theinertia of the particle.For this case, we have the Langevin equation m ¨ x + γ ˙ x + kx = η ( t ) , (31)with the heat given by Q [ x ( τ )] = 12 m ( v t − v ) + 12 k ( x t − x ) . (32)Following the same steps of the last case. In order to solve for P ( Q, t ), we need to solvethe integrals: P ( Q, τ ) = Z d Γ t Z d Γ P (Γ ) Z dλ π e iλ ( Q − m ( v t − v )+ k ( x t − x ) ) Z v ,v t ,x t ,x Dx e A [ x ( t )] , (33) he Heat Distribution of the Underdamped Langevin Equation. Figure 2.
Heat distribution for the case 2. With parameters m, κ, T, γ = 1 plotted in3 different times t=1 (red dashed line), t=2, (pink long dashed line), and t=3 (purplesolid line). where now, the stochastic action will be A [ x ( t )] = − γT Z τ ( m ¨ x + γ ˙ x + kx ) dt. (34)Notice that the action is quadratic, so the path integral will be exact. We solve the pathintegral in Appendix B, showing the dependence on the boundary terms. Defining theboundary vector ~b = ( v t , v , x t , x ), we are now left with the integral P ( Q, τ ) = N ′ Z dλ π e iλQ Z d ~b P ( v , x ) exp (cid:16) − ~b D ( λ ) ~b T (cid:17) (35)where D ( λ ) is a matrix formed by the result of the path integral and the constraintgiven by the delta, which gives the dependence in λ .The initial probability P ( v , x ) isthe thermalized Boltzmann distribution, with Hamiltonian H = m p + k x , p = mv .We show the matrix D ( λ ) in the Appendix B. Note that, the use of this notation is forclarification of the steps. All the integration’s in x , v , x t , v t can be rewritten in a fourdimensional integral over ~b .After the integration in the boundary terms, we find P ( Q, τ ) = N ′ I ′ ( Q, τ ) = N ′ Z dλ e iλQ q k m λ + α λ + α (36)where N ′ , α , α give the dependence on τ and are given in the Appendix B. Theabove integral, as far as we know, cannot be solved analytically. However, we can he Heat Distribution of the Underdamped Langevin Equation. Figure 3.
Heat Distribution for the Case 3. The values of the parameters are γ = 2 , k = 1 / , m, T = 1. find exact results for this distribution through numerical integration [46]. The shape ofthe distribution in (36) is given in figure 3. Moreover, it can be shown easily that thisresult recovers case 1 in the limit k = 0.Recently, with has been derived the heat distribution of this case in the equilibriumregime [34], that is, in the limit τ → ∞ . The found distribution here, agrees with thisresult in the proper limit, and we can show analytically. In the limit τ → ∞ the (36)becomes P ( Q, ∞ ) = 12 π Z dλ e iλQ q (1 + λ T ) = 12 T exp − | Q | T ! . (37)From the asymptotic distribution above, it is easy to check the normalization condition.The asymptotic distribution have a symmetric dependence on Q , which can be view inthe curve τ = 10 of figure 3.There is an unexpected result in the above derivation. In order to get the Gaussianintegral convergence, we need that the parameters satisfy γ > √ km , this is a timescaleconstraint, since τ − γ = γ/m and τ − k = q k/m . This is a mathematical necessity in thesteps made here. Writing in terms of the time scale, the constraint become τ γ < τ k . (38) he Heat Distribution of the Underdamped Langevin Equation.
5. Discussion and Conclusion
We have derived the heat distribution of the massive Brownian particle for three cases:the free particle, the linear potential, and the quadratic potential. We call these casesrespectively, as cases 1, 2, and 3. Using the path integral formalism we are able to findexact results for these heat distributions.For case 1, we have a symmetric distribution in the argument Q , this means thatthe particle has no tendency to gain or to lose energy from the heat bath. Moreover,due to this symmetry, we have that P ( Q ) = P ( − Q ). It is interesting to note thatthis case has a mathematical equivalence with the overdamped particle in a quadraticpotential [10], this equivalence comes from the fact that we can write the second-orderstochastic differential equation for x ( t ) as a first-order stochastic differential equationin v ( t ). There is a singularity in Q = 0 which comes from the Bessel function K. Wepointed out that this singularity is in agreement with the overdamped limit, where wefind a delta peak in Q = 0. Despite this singularity, the distribution will be properlynormalized as already shown in [10]. Unexpectedly, the distribution we derived doesnot have a direct correspondence with the overdamped case. The results of the heatdistribution are different if we take the overdamped limit before in the dynamics [44].This non correspondence is in current study in the literature [30, 31, 32]. We pointedout, for the case discussed here, that there is need of further investigations on thisproblem.In case 2, with a linear potential, we break the symmetry found previously of case1. Qualitatively, we can see in figure 1 the behavior of the distribution. The distribution P ( Q ) starts with a peak located very nearly zero, however, as time pass the distributionstarts to spread towards the negative values of Q . This means that it will be moreprobable for the particle to lose energy from the bath as time pass, as the work done bythe external force is transformed into heat that is dumped in the thermal reservoir. Areasonable interpretation of case 2 is a charged Brownian particle inside a thermal bathcontained between the plates of a capacitor, and the force as the one generated by theuniform electric field. It is straightforward to observe that the particle loses energy tothe bath due to the dragging action of the electric field. In this case, we were not ableto solve the integration analytically. However, we managed to get a numerically exactresult through numerical integration of the integral in (30). Coincidentally, this casehas the same structure of the overdamped “parabola sliding system” [12]. The deriveddistributions, for both systems, are written in terms of integrals in λ that are closelyrelated. Comparing cases 1 and 2, with the respective similar overdamped systems, putin evidence an analogy between the underdamped and the overdamped case for thesesystems. The drift term in the underdamped cases fulfills the role of the harmonicforce for the overdamped cases, yielding results which are mathematically similar. Thiscorrespondence does not hold for the next case.For case 3, the damped harmonic oscillator case, we found a distributionqualitatively similar to case 1. Even though we cannot write the distribution in terms he Heat Distribution of the Underdamped Langevin Equation. Q , with exponential tail behavior. The big difference is a fourth-order polynomial in the denominator of (36), while in (21) we have a second-orderpolynomial. Plotting the distribution of case 3, we can see that the distribution issymmetric in Q , giving a reversible distribution, that is P ( Q ) = P ( − Q ). This long timesasymptotic behavior can be obtained analytically as t → ∞ , was recently in evidence inthe literature where the exponential dependence on the modulus of Q was found [34].indeed, this result we were able to recover, using the path integral technique. In addition,we notice that this distribution has a parallel behavior with the heat distribution of twooverdamped particles in contact with two heat bath [13]. This correspondence comesfrom the similar dependence of the exchanged heat in the four boundary terms of therespective two cases. Differently from the previous cases, the correspondence is not inthe Langevin equation, it is in the similar functional form of the heat in both cases.Furthermore, for the heat distribution to be derived, we need to use the mathematicalrelation γ > √ km , which is a surprising result, since it was naively expected that thedistribution would be valid for any values of the parameters. This constraint can beviewed as a time-scale constraint, and is left as an open question for future work.Comparing the three cases, since they differ just by the qualitatively differentapplied forces, it is logical to find some common features between them. All the threeheat distributions, (21), (30), and (36), can be written in terms of an oscillatory integral,with oscillatory exponential dependence in Q and a polynomial in the denominator ofthe integral inside a square root. For case 1, this gives an exact Bessel function K, whilein the two cases we obtain a more general integral, which prevents us to write it in termsof known functions. These unknown functions have very similar forms to the integraldefinition of the Bessel function case. Moreover, cases 2 and 3 are equivalent to the case1 in the respectively limits κ = 0 and k = 0. This is a necessary consistency test for ourresults.In summary, we have derived the heat distribution in the underdamped regime forthree different cases: the free particle, the constant force, and the harmonic oscillator.We review the Stochastic Thermodynamics of such systems and through the use of thepath integral formalism of the stochastic processes. We were able to find exact resultsfor the heat distributions. The work done here helps to enlarge the body of the literatureof Stochastic Thermodynamics, giving new results for very studied Brownian systems.In particular, the heat of the original Langevin equation has never been studied indetail, and the heat distribution of the harmonic oscillator was only studied for theequilibrium regime [34], or the overdamped regime [10], as far as we know. Thus, ourfindings generalize these cases. Moreover, the harmonic case is important because itappears in well-known experiments as the Carnot Brownian engine [26] and the StirlingBrownian engine [47]. New directions can come from the present work, generalizing thelist of results in heat distributions of Stochastic Thermodynamics. We point out thatusing the same methods used herein it is possible to solve for the heat in the case ofexternal drive force, and sliding parabola in the underdamped regime, which could be he Heat Distribution of the Underdamped Langevin Equation. Acknowledgments
This work is supported by the Brazilian agencies CAPES and CNPq. PVP would like tothank CNPq for his current fellowship. RA was partially supported by a PhD Fellowshipfrom FAPERJ. This study was financed in part by Coordena¸c˜ao de Aperfei¸coamento dePessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.
Appendix A. Calculations Case 2
Appendix A.1. Path Integral A [ x ( t )] = Z τ ( m ¨ x + γ ˙ x − κ ) dt (A.1)The extremization of this action will give the solution x c ( t ), δ A = 0 → m x (4) c ( t ) = γ ¨ x c ( t ) , (A.2)where the superscript (4) denote 4-th order time derivatives. Solving the abovedifferential equation we find x c ( t ) = c m e γtm γ + c m e − γtm γ + c t + c (A.3)where the c ′ s are arbitrary constant, which needs to be fixed by the equations: x c (0) = x , x c ( τ ) = x t , ˙ x c (0) = v , ˙ x c ( τ ) = v t . The action evaluated in x c will be A [ x c ] = γ c τ − γc κτ + κ τ + 4 c c m (cid:16) e γτm − (cid:17) + (A.4)+ 2 c m γ (cid:16) e γτm − (cid:17) (cid:16) − κ + c m (cid:16) e γτm + 1 (cid:17)(cid:17) (A.5)where c = γ m v t − v e γτm − γ ( τ ( v + v t ) + 2 x − x t ) γτ + e γτm ( γτ − m ) + 2 m ! , (A.6) c = m ( v + v t ) (cid:16) e γτm − (cid:17) + γ ( x − x t ) (cid:16) e γτm + 1 (cid:17) − γτ + e γτm (2 m − γτ ) − m . (A.7)Substituting these constants we have the function f , which is written in (A.9).Following the steps explained in section 3 we have the normalization factor NN = γ πT √ me γτ m r γτ sinh (cid:16) γτm (cid:17) − m cosh (cid:16) γτm (cid:17) + 2 m . (A.8)This term correct normalized the path integral for the action in (A.5). he Heat Distribution of the Underdamped Langevin Equation. Appendix A.2. Omitted Terms
We give the terms omitted in the calculation of the heat distribution for the case 2. Thefunction f in (27) is given by f ( v t , v , x t , x ) = − γT (cid:16) γτ coth (cid:16) γτ m (cid:17) − m (cid:17) × " m ( γ ( v + v t ) − κ )(2 κτ + 4 m ( v − v t ) + γτ ( v + v t ) + 4 γ ( x − x t )) ++ γ coth (cid:18) γτ m (cid:19) (cid:16) m ( v − v t ) + 2 mτ ( v − v t )( γ ( v + v t ) − κ )+ − γmτ ( v − v t ) coth (cid:18) γτ m (cid:19) − κτ + γx − γx t ) (cid:19)(cid:21) (A.9)It is clear, that the dependence on the Boundary terms are a second order polynomial.The function f in (30) is given by f ( λ ) = 1 γ (cid:16) cosh (cid:16) γτm (cid:17) + (2 λ T + 1) sinh (cid:16) γτm (cid:17)(cid:17) × (A.10) (cid:20) κ λ T (cid:18) m (cid:18) cosh (cid:18) γτm (cid:19) − (cid:19) − γτ sinh (cid:18) γτm (cid:19)(cid:19) ++2 iκ λ T (cid:18) γτ sinh (cid:18) γτm (cid:19) − m cosh (cid:18) γτm (cid:19) + 2 m (cid:19) + − iκ λ (cid:18) ( m − γτ ) sinh (cid:18) γτm (cid:19) + (2 m − γτ ) cosh (cid:18) γτm (cid:19) − m (cid:19) ++ κ λ T (cid:18) ( m − γτ ) sinh (cid:18) γτm (cid:19) + (2 m − γτ ) cosh (cid:18) γτm (cid:19) − m (cid:19)(cid:21) . Appendix B. Calculations Case 3
Appendix B.1. Path Integral
Similarly, we have A [ x ( t )] = − γT Z τ ( m ¨ x + γ ˙ x + kx ) dt. (B.1)Once again, we extremize the stochastic action, obtaining, m x (4) ( t ) + (2 km − γ )¨ x ( t ) + k x ( t ) = 0 . (B.2)The extreme solution is, x c ( t ) = d exp − Ψ + t m ! + d exp − Ψ − t m ! + d exp Ψ − t m ! + d exp Ψ + t m ! , (B.3)with Ψ + = γ + √ γ − km and Ψ − = γ − √ γ − km . Just like case 2, we will usethe initial conditions x c (0) = x , x c ( τ ) = x t , ˙ x c (0) = v and ˙ x c ( τ ) = v t to compute theconstants d ’s. The extreme action reads, A [ x c ] = 14 mT " γ d Ψ − (cid:18) e Ψ − τm − (cid:19) + 8 d d k m (cid:16) e γτm − (cid:17) + γ d Ψ + (cid:18) e Ψ+ τm − (cid:19) (B.4) he Heat Distribution of the Underdamped Langevin Equation. d = e − Ψ+ τ m γ e ( − γ ) τ m (cid:16) x Ψ + + 2 mv (cid:17) − m e Ψ+ τ m (cid:16) v Ψ + + 2 kx (cid:17) + e − Ψ − τ m (cid:16) mv q γ − km − γx Ψ + + 4 kmx (cid:17) − m e √ γ − kmτm (cid:16) v t Ψ + + 2 kx t (cid:17) ++ e Ψ+ τm (cid:16) mv t q γ − km − γx t Ψ + + 4 kmx t (cid:17) + γ (cid:16) x t Ψ + + 2 mv t (cid:17)! (B.5) d = e − τ √ γ − kmm γe − ( −− γ ) τ m ( x t Ψ − + 2 mv t ) + γ ( x Ψ − + 2 mv ) + − m e − Ψ − τ m ( v t Ψ − + 2 kx t ) + e − Ψ − τm (4 kmx − mv q γ − km − γx Ψ − ) + e Ψ+ τm (4 kmx t − mv t q γ − km − γx t Ψ − ) − m e τ √ γ − kmm ( v Ψ − + 2 kx ) ! (B.6)And, in both constants, the factor ∆ is,∆ = − km + (4 km − γ ) cosh (cid:18) γτm (cid:19) + γ cosh τ √ γ − kmm ! . (B.7)Although in the extreme action we have long constants terms, due to the large setof parameters involved in our specific model, we can rewrite A [ x c ] in a bilinear formin terms of vector ~b = ( v t , v , x t , x ). So, accordingly to (13) in the main text, thenormalized path integral is,exp (cid:16) − γT A [ v c , x c ] (cid:17)R dv t dx t exp (cid:16) − γT A [ v c , x c ] (cid:17) = N ′ exp (cid:18) − D~b M ~b T (cid:19) (B.8)where N ′ = q km (4 km − γ ) e γτ m √ πT s γ cosh (cid:18) τ √ γ − kmm (cid:19) + (4 km − γ ) cosh (cid:16) γτm (cid:17) − km (B.9)and M = A A A A A + m T − A − A A A A + k T (B.10)The terms A i ’s in the matrix M contains the dependence on the constants of theproblem. These terms are A = m T ∆ − γ q γ − km sinh τ √ γ − kmm ! + γ − cosh τ √ γ − kmm !! ++ (cid:16) γ − km (cid:17) (cid:18) cosh (cid:18) γτm (cid:19) + sinh (cid:18) γτm (cid:19)(cid:19) + 4 km (cid:19) , (B.11) he Heat Distribution of the Underdamped Langevin Equation. A = km A + kγ √ γ − km sinh (cid:18) τ √ γ − kmm (cid:19) T ∆ , (B.12) A = mT ∆ (cid:16) km − γ (cid:17) sinh (cid:18) γτ m (cid:19) cosh τ √ γ − km m ! ++ γ q γ − km cosh (cid:18) γτ m (cid:19) sinh τ √ γ − km m !! , (B.13) A = − γkm sinh (cid:18) τ √ γ − km m (cid:19) T ∆ , (B.14) A = 2 km √ γ − km sinh (cid:16) γτ m (cid:17) sinh (cid:18) τ √ γ − km m (cid:19) T ∆ , (B.15) A = km A − kγ √ γ − km cosh (cid:16) γτ m (cid:17) sinh (cid:18) τ √ γ − km m (cid:19) T ∆ . (B.16) Appendix B.2. Omitted Terms
The matrix D ( λ ) is just the sum of M (B.10) and the λ dependence given by the delta,which is just a diagonal matrix. We have in (35) D ( λ ) = M + iλ diag ( − m, m, − k, k ) , (B.17)where diag means that we have a 4 dimensional diagonal matrix, with elements givenin the argument.The normalization constant in the probability (36) N ′ = km q ( γ − km ) e γτ m √ πT s km − γ cosh (cid:18) τ √ γ − kmm (cid:19) − (4 km − γ ) cosh (cid:16) γτm (cid:17) , (B.18)whereas the parameters α , are: α = k m T (4 km − γ ) sinh (cid:16) γτm (cid:17) γ cosh (cid:18) τ √ γ − kmm (cid:19) + (4 km − γ ) cosh (cid:16) γτm (cid:17) − km + 1 , (B.19) α = k m (4 km − γ ) e γτm T (cid:18) γ cosh (cid:18) τ √ γ − kmm (cid:19) + (4 km − γ ) cosh (cid:16) γτm (cid:17) − km (cid:19) . (B.20) References [1] Ciliberto S 2017
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