Derivation of the Langevin Equation from the Microcanonical Ensemble

When writing down a Langevin equation for the time evolution of a “system” in contact with a thermal bath, one typically makes the implicit (and often tacit) assumption that the thermal environment is in equilibrium at all times. Here, we take this assumption as a starting point to formulate the problem of a system evolving in contact with a thermal bath from the perspective of the bath, which, since it is in equilibrium, can be described by the microcanonical ensemble. We show that the microcanonical ensemble of the bath, together with the Hamiltonian equations of motion for all the constituents of the bath and system together, give rise to a Langevin equation for the system evolution alone. The friction coefficient turns out to be given in terms of auto-correlation functions of the interaction forces between the bath particles and the system, and the Einstein relation is recovered. Moreover, the connection to the Fokker–Planck equation is established.


Introduction
The Langevin equation [1][2][3][4][5][6] is a well-established and extremely successful model for describing how a system evolves under the combined influence of deterministic and rapidly fluctuating ("random") forces.Typically, the random forces and their stochastic nature are the result of many (microscopic) degrees of freedom, which change extremely quickly compared to the "slow" degrees of freedom of the system.A paradigmatic application is Brownian motion [2], i.e., the diffusive motion of a small particle in a fluid or gas, as schematically illustrated in Figure 1 (the Langevin equation for this setup is explicitly stated in (55)).Generally, a macroscopic thermodynamic system plays the role of a "thermal bath" or "environment" for a small system of interest.A common standard assumption when investigating the effective stochastic dynamics of such systems is that the thermal bath remains in equilibrium at all times [6].
This equilibrium assumption is justified by an extreme separation of time scales.The bath possesses a fast intrinsic time scale τ on which perturbations away from equilibrium decay rapidly (typically, τ is related to the molecular collision time, which is about 10 −13 s in water).The dynamics of the system occur on time scales much slower than this characteristic time scale of the bath.For instance, for a colloidal particle, the fastest time scale is the velocity relaxation time due to viscous friction, which is several orders of magnitude larger than τ (it is about 10 −8 s for a colloidal sphere of radius 0.1 µm in water).Its (diffusive) spatial motion occurs on even slower time scales (about ms).Moreover, in typical experiments, external time-dependent variations in forces, etc., exerted on the system do not directly interfere with the bath and vary at times much larger than τ.Note that we completely disregard that perturbations might create slow collective "degrees of freedom" in the bath.Even though this is known to be the case in dense environments, the modifications of the system dynamics due to such slow collective modes in the bath seem to be of little practical relevance for typical systems usually modeled by Langevin equations.Put differently, the theory that we are going to develop here is a proper description for systems for which such slow bath modes do not have observable consequences.
Hence, at any instance of slow "system time", i.e., at any time resolution ∆t of interest in typical experiments that capture the system behavior (with ∆t ≫ τ being microscopically large but macroscopically small), the bath is in an equilibrium state with a well-defined energy.Energy changes occur only via processes connected to changes in the system state, e.g., via a displacement of the colloidal particle.For the fast microscopic degrees of freedom of the bath.such processes, occurring over ∆t time scales, are quasi-statically slow transformations from a given equilibrium state to a new one.Since the bath is isolated from the rest of the world, each of these equilibrium states can be characterized by the microcanonical ensemble for the current value of bath energy.
Based on a classical Hamiltonian description of the system and bath together (Section 2), and the representation of the bath state by the microcanonical ensemble, we analyze the system dynamics on ∆t time scales.We show that individual ∆t steps are statistically independent (Section 3.1), implying that the system dynamics are Markovian on ∆t time scales.For individual time steps ∆t, we calculate averages and correlations of changes in the system state (Section 3.2).In linear order ∆t, these moments turn out to be identical to those generated by a standard Langevin equation for the system dynamics alone (Section 3.3).Hence, we demonstrate that the standard Langevin equation represents an "effective" evolution equation for a system in contact with a bath, both together described on the microscopic level by a classical Hamiltonian.We also derive from the Hamiltonian description that the proper rules of stochastic calculus [3,4] for the effective Langevin equation are given by the Stratonovich interpretation (Section 3.4).

Model
We consider the setup schematically illustrated in Figure 1.A macroscopically large thermodynamic system plays the role of a thermal bath or environment for a (small) system of interest.We denote all parameters and coordinates, etc., related to the bath by small letters, and those specifying the system by capital letters; Greek indices denote particle numbers and Latin indices denote components of three-dimensional (coordinate) vectors.We collect all bath degrees of freedom in the super-vector ϕ = (p, q) = (p 1 , p 2 , . . . ,p n , q 1 , q 2 , . . . ,q n ) and all system degrees of freedom in Φ = (P, Q) = (P 1 , P 2 , . . ., P N , Q 1 , Q 2 , . . ., Q N ).We assume that N ≪ n.The total Hamiltonian of the bath and system together is with The interactions between the bath particles in the bulk are captured in U bulk (|q µ − q ν |), and the interaction between bath particles and the system is captured in U int (|q µ − Q ν |).Both depend on particle-particle distances.We combine H bulk (ϕ) and H int (ϕ, Q) into the bath Hamiltonian H bath (ϕ, Q) because, for the effective system dynamics that we are aiming at, the system energy should be determined by H sys (Φ, λ) alone without reference to bath degrees of freedom.The potential U(Q, λ) represents the potential forces applied to the system by external means; they affect only positional degrees of freedom (not momenta).The potential is, moreover, allowed to depend on time via the protocol λ = λ(t), varying noticeably only on time scales much larger than τ.
thermal bath (isolated) system (e.g., colloidal particle) Figure 1.The setup considered here consists of a thermal bath and a "system" in contact with that bath, e.g., a colloidal particle suspended in water at room temperature.The bath is isolated from the rest of the universe and can interact only with the system.The system can be manipulated by external means (e.g., via an externally applied potential), which, however, do not affect the bath directly.(image credit: Jasper Eichhorn).

Dynamics
The Hamilton equations of motion are [7] where the partial derivative symbols with respect to a vector quantity denote the corresponding gradient vector.Following standard convention in physics, we use the same symbols for the solutions of these equations as for the coordinates in phase space (ϕ, Φ), just endowing them with a time argument, which we write here as an index; if useful, we will also explicitly state the dependence on the initial values.The formal solutions for the various components p = (p 1 , p 2 , . . ., p n ), q = (q 1 , q 2 , . . ., q n ) and P = (P 1 , P 2 , . . ., P N ), for an initial configuration (ϕ 0 , Φ 0 ) at time t 0 .Here, the notation means that we evaluate the derivative of H bath (ϕ, Q) with respect to q (this is the 3n-dimensional gradient vector) and evaluate it along the trajectory (ϕ t ′ , Q t ′ ), and likewise for similar expressions.
According to the time scale separation discussed above, (q t , p t ) vary extremely rapidly on the molecular collision time scale τ, while (Q t , P t ) are virtually constant on that scale but change quickly on macroscopic time scales.The system degrees of freedom (Q t , P t ) therefore evolve on a mesoscopic time scale ∆t with τ ≪ ∆t ≪ (macroscopic time scale).For the time evolution over such a time interval ∆t, we can write with Q denoting the system configuration at time t.A similar expansion in powers of ∆t is not possible for q t (or p t ) due to its rapid fluctuations.Likewise, P t cannot be directly expanded in ∆t because it is governed by interactions with the fast bath degrees of freedom (see Equation (3c)).

Statistical Mechanics
The total energy of the bath and system together, is conserved by the dynamics (2) for fixed λ.In the general case of external time-dependent protocols λ = λ(t), the energy is not conserved.However, only slow changes in λ(t) over mesoscopic time intervals ∆t ≫ τ are permitted (and relevant experimentally).From the perspective of the bath, variations in λ are thus quasi-statically slow, such that the bath has a well-defined equilibrium energy within each time interval ∆t.This energy is determined by the current values of Φ, Note that if all parameters (ϕ, Φ) remain fixed and only λ is varied, the total energy is changed via the contribution in the system potential, while the bath energy remains unchanged (and therefore does not directly depend on λ).According to our equilibrium assumption, the bath can be described by the microcanonical ensemble, which is consistent with the current values of Φ and λ.The corresponding microcanonical partition function Ω(E) is given by [8] Ω According to standard statistical mechanics [8], the microcanonical density is and the temperature T of the bath is defined as (k B is Boltzmann's constant)

Effective System Dynamics
Starting from an initial state of the bath and system together, the specific evolution of the system state Φ according to (2c), (2d) is affected by the rapidly fluctuating bath degrees of freedom, uniquely emerging from their initial state (and the one of the system).Different initial states of the bath thus lead to different "realizations" of the system dynamics, even if they start from the same initial configuration.Since we do not know the initial state of the bath exactly, but only that it is consistent with the microcanonical density (8), we can write the transition probability density for the system to evolve from Φ = (P, Q) to Φ ′ = (P ′ , Q ′ ) during a mesoscopic time interval ∆t as where we denote the initial state of the bath at the beginning of the time interval by the unprimed variable ϕ.This probability density gives rise to the statistical properties of "small displacements" ∆Φ = Φ ′ − Φ of the system degrees of freedom.Their analysis is the central aim of this paper.Before evaluating averages and (co-)variances in Section 3.2, we start in Section 3.1 with demonstrating that the system displacements within successive time steps ∆t are independent.

Markov Property
We consider a configuration (ϕ, Φ) of the bath and system together, which, from time t, evolves for a duration 2∆t according to the full microscopic dynamics (2).Generalizing (10), the probability density for finding the system in configuration Φ ′ at time t + ∆t and in configuration Φ ′′ at time t + 2∆t when it started at Φ at time t is We will show in the following that p(Φ ′′ , t + 2∆t; Φ ′ , t + ∆t|Φ, t) fulfills the Markov property where in analogy to (10) (we skip the energy subscript at p for notational convenience).The energy E ′ in (13b) is given by the energy balance (6) for the primed variables, and with λ = λ(t + ∆t).

Starting from the expression (11), we first use the composition property of solutions to the Hamiltonian equations of motion to write
. Then, we insert unity into the integral, expressed in the form dϕ ′ δ ϕ ′ − ϕ ∆t (ϕ, Φ) .With the presence of this delta function in the integrand, we can replace ϕ ∆t (ϕ, Φ) everywhere by ϕ ′ , and likewise for Φ ′ due to the delta function δ Φ ′ − Φ ∆t (ϕ, Φ) .These steps turn (11) into exchanging the order of integration in the second equality.Here, the second line is the probability density p(ϕ ′ , Φ ′ , t + ∆t|Φ, t) for the total system to be in the state (ϕ ′ , Φ ′ ) after evolving for time ∆t, given that the system is in state Φ at time t.Hence, So far, the calculations are exact.Now, however, we have to make use of our central assumption that the bath is in equilibrium at any time point t, t + ∆t, t + 2∆t, etc., with a bath energy that is consistent with the current state of the system.This implies that the probability density of bath configurations ϕ ′ depends only on the system state at the same time point, but not on its state at earlier times (this is used in the second step below), and that it is given by the microcanonical density (used in the third step, cf.Equation ( 8)).We can thus write Plugging this relation into (15), we obtain In the second equality, we used the relation (13b), completing the proof of the Markov property ( 12) for the effective system dynamics on ∆t time scales.

Average System Behavior
The main idea is to calculate from (10) the average displacements ⟨Φ ′ − Φ⟩ = ⟨∆Φ⟩ and various (co-)variances ⟨∆Φ ∆Φ⟩, in linear order in ∆t, and to assess which kind of stochastic evolution equation for the system dynamics alone would produce the same moments in leading order ∆t.According to our result from the previous section, it is sufficient to consider any such time interval ∆t because the system dynamics are Markovian.
For clarity, we consider just one system particle such that Φ = (P, Q) = (P, Q) with P = P = (P 1 , P 2 , P 3 ) and We then have to evaluate (i, j = 1, 2, 3) to lowest order in ∆t.Note that the averages in (18) are over the final configuration Φ ′ only, conditioned on a fixed (but arbitrary) initial value Φ.

Evaluation of ⟨∆Q i ⟩
We use the expression (10) for p E (Φ ′ |Φ) in (18a): Since (Q i ) ∆t (ϕ, Φ) is a slowly varying function of t on ∆t time intervals, we can use the expansion from (4) and obtain The linear order result is therefore, as expected, We start again by using the expression (10) for p E (Φ ′ |Φ), where, in the last line, we inserted the formal solution from (3c).The system contribution (second term in the brackets) is a slowly varying function of time on ∆t scales.Defining the external force on the system, and expanding ).Using this expansion, we can perform the time integral and the microcanonical average for the system contribution.We obtain for ( 22) A similar expansion procedure is not achievable for the interaction part because just superimposes a slow drift over the rapid fluctuations.Nevertheless, in order to analyze this term, we can start by evaluating the "zeroth-order" effect by approximating Q t ′ with its initial value, Q t = Q.The interaction term is then The represents the momentary, average net force (component i) that the bath exerts on an immobile system particle.For physical reasons, we expect this force to vanish, as there should be no net force on the particle from the thermal environment.In Appendix A, we prove that this is indeed the case.
The question now arises as to whether there will be a net ∆t contribution from when considering the next order Q t ′ ≈ Q + (P/M)(t ′ − t) in the particle displacement.In this case, the particle moves (slowly) relative to the bath and we indeed expect a frictional contribution to show up, which, in lowest order, should be proportional to (P/M)∆t (see also the illuminating discussion in Section 15.5 of [8]).Since we cannot expand in t ′ , we extract the leading contribution by a partial integration in the time integral.Moreover, from a physical perspective, it will be instructive to transform to coordinates co-moving with the system particle.We will perform the calculation with and without the transformation to the co-moving frame, which will result in two alternative (but equivalent) expressions for the prefactor of (P/M)∆t.

Co-Moving Frame
In order to switch to the co-moving frame, we introduce for all µ = 1, 2, . . ., n the time-dependent coordinate transformation (for fixed initial time t at which the bath and system are in the state (ϕ t , Φ t ) = (ϕ, Φ)) Differences between bath particle coordinates are invariant under this transformation, implying that Differences between bath particle coordinates and the system coordinate transform according to Here, we use the formal solutions (3b) and (3d) (see also (4)).We moreover note that the solution (3a) for p µ , i.e., (p µ , contains the transformed via its dependence on the differences q µ − Q, and analogously for similar expressions.The transformation to the co-moving frame can therefore be seen as a time-dependent shift q → q(t ′ ) in the initial q value.Since the integral dq in ( 27) is over all possible q values, and since dq = d q(t ′ ), this is the same for all transformations q(t ′ ), independent of the time point t ′ .Within the integral, q(t ′ ) therefore plays the role of a new initial value, i.e., under the integral, we can drop the time argument in q(t ′ ).In the following calculation (second equal sign below), we make this explicit by replacing q(t ′ ) with the symbol q, which we used to indicate the initial values at starting time t all along.Using the transformation (28), we apply (29) to rewrite the interaction part (27) as The main effect of the transformation to the co-moving frame is thus to move the time dependence in Q t ′ from the interaction force outside the δ function to the interaction Hamiltonian inside the δ function.
In order to extract the leading order in ∆t, we introduce ] in the time integral and perform a partial integration (using the "more natural" insertion ] would lead to non-vanishing boundary terms after partial integration): The first line in the last equality is the boundary term from the partial integration.In addition, the partial integration produces two new terms (second and third summand) due to the (slow) time dependence in the interaction Hamiltonian within the δ function and the (fast) one in the interaction force outside the δ function.
Next, we transform back to the original coordinates in the laboratory-fixed frame (and use q t = q, Q t = Q again): The first line corresponds to the zeroth-order contribution discussed above (see Equations (25) and ( 26)) and is proven in Appendix A to vanish for homogeneous, translationalinvariant environments (in other cases, this term would result in a net force from a so-called "potential of mean force" [9]).Likewise, we show in Appendix A that phase-space averages of the type like the one in the last line vanish in zeroth order ∆t, for which , implying that they contribute only in O(∆t 2 ).Hence, the only contribution in linear order ∆t comes from the term in the second and third line when approximating Q t ′ ≈ Q.The prime at the δ function denotes the derivative with respect to its argument, or, equivalently, with respect to E. Using the definition of temperature (9), we prove in Appendix B that, for "non-extensive" averages (like averages over short-ranged interactions between the system particle and bath), we are allowed to replace δ ′ (. ..) with 1 k B T δ(. ..).We finally find that Laboratory Frame We can, of course, perform exactly the same partial time integration as in the calculation above without transforming to the co-moving frame.Using again the results (A7) from Appendix A, which imply that certain terms do not contribute in linear order ∆t (in the third and fourth step below), the main calculation steps are Finally, The O(∆t) Contribution in ⟨∆P i ⟩ Summarizing what we found for ⟨∆P i ⟩, the contribution to linear order in ∆t is Here, we introduce the friction coefficient γ ij , given by the two alternative expressions from (33) and (35): We remark that the equivalence of 1 verified directly; see Appendix C. It is obvious from the second line that γ ij is symmetric under exchange of the indices i,j, a property that is then also valid for the auto-correlation in the first line.The first expression represents a time integral over the correlation of the interaction forces, averaged over the various dynamical evolutions of the bath particles in the microcanonical ensemble.By shifting the integration variable t ′ to the new variable t = t ′ − t and by defining qt = q t+t (such that q0 = q t = q), the time integration runs over the interval [0, ∆t], where we drop the tilde symbol to simplify notation.Since we assume that perturbations in the bath decay on the characteristic time scale τ, we expect the correlations to also decay on this time scale, such that vanishes for times t ≫ τ.As a consequence, we can extend the upper integration limit ∆t ≫ τ to infinity.Moreover, the term t/∆t in the integral is effectively O(τ/∆t) and thus negligibly small, as can be verified by rescaling time according to t = t/τ (to make t ∼ O(1) when t ∼ O(τ)).Hence, our expression for the friction tensor becomes This is exactly the same result as obtained from projection operator techniques; see, e.g., Chapter 11 in [2].

Evaluation of ⟨∆Q i ∆Q j ⟩
The evaluation of ⟨∆Q i ∆Q j ⟩ proceeds along similar lines as the one of ⟨∆Q i ⟩.We use (10) in (18c) to write For (Q i ) ∆t (ϕ, Φ), we can again use the expansion 4)) so that we obtain As expected, ⟨∆Q i ∆Q j ⟩ does not contribute in linear order in ∆t.

Evaluation of ⟨∆P i ∆P j ⟩
In order to evaluate ⟨∆P i ∆P j ⟩, we proceed analogously.We start from (18c) and plug in the transition probability (10) and then the formal solution (3c): The terms involving H sys (Φ t ′ , λ t ′ ) represent the external forces (23) exerted on the system.Recalling that these vary slowly in time on ∆t intervals, we obtain The microcanonical averages appearing in the first two lines are the average net forces of the bath on an immobile system particle, as in (26), and are proven to vanish in Appendix A, cf.Equation (A7a).We thus obtain In order to find out if and how this double time integral is related to the friction tensor (37), our aim is to rewrite it as a single time integral.We first split up the square integration domain into two triangular domains and then change the order of integration in one of the domains: The last equality is obtained by renaming the integration variables t ′ ↔ t ′′ in the first term.
In the next step, we shift the integration interval in the inner integral by introducing the new variable t = t ′′ − (t ′ − t): The correlations inside the integrals are now expressed in terms of the bath states at time t ′ and at the later time t ′ + ( t − t).Due to our central assumption that the bath is in equilibrium, these correlations are functions of the time difference only, i.e., they depend on t − t but not t ′ .We are thus free to choose any time point for t ′ , and we fix it to the initial time t of the considered time interval ∆t (remember that the initial bath state is Exploiting the i ↔ j symmetry of the correlations (cf.Equation (37)), the two terms can now be combined into one.Moreover, since the correlations no longer depend on t ′ , we can perform the t ′ integral by exchanging the order of integration: In linear order in ∆t, we therefore find Finally, with the definition (37) of the friction coefficient, we can write this result in the suggestive form 3.2.5.Evaluation of ⟨∆Q i ∆P j ⟩ Finally, we quickly verify that ⟨∆Q i ∆P j ⟩ has a quadratic (and higher) contribution in ∆t.We start from (18e), plug in the expression (10) for p E (Φ ′ |Φ) and use the formal solutions (3c), (3d) to assess the various contributions: Expanding the slowly varying system position (Q i ) t ′ at lowest order ∆t, we can approximate t+∆t t dt ′ (P i ) t ′ M by P i M ∆t (cf.also (20)).We are then basically left with an average over ∆P j , for which we can re-use the results from Section 3.2.2.We thus obtain

Summary
Collecting all contributions that are linear in ∆t, we have with the tensorial friction coefficient All other averages or correlations are at least O(∆t 2 ).
It is straightforward to verify that a standard Langevin equation of the form with mutually independent, unbiased, δ-correlated white noise processes ξ j (t), generates identical average "displacements" and correlations in linear order ∆t (remember that the averages in (53) are conditioned on the initial configuration at the beginning of the time step ∆t).Moreover, the diffusion coefficient quantifying the mean square displacement of the spatial coordinates Q is obeying Einstein's fluctuation-dissipation relation [2][3][4][5].
Strictly speaking, we did not demonstrate that the fluctuations behind the averages (53) are Gaussian.However, with the huge separation of time scales τ ≪ ∆t, we can assume that there exists an "intermediate" time scale ∆t, with τ ≪ ∆t ≪ ∆t.It is still much larger than τ, so all the above results apply to displacements over ∆t as well, but it is considerably smaller than ∆t, so displacements over ∆t are the result of many independent and identically distributed displacements over ∆t.According to the central limit theorem, the resulting dynamics on ∆t scales are then Gaussian-distributed with the moments and correlations given above in (53).

Rules of Calculus
Since we find that the statistical properties of the system momentum are generated by a white noise source, the question about the proper rules of (stochastic) calculus naturally arises, eminent in the procedure of calculating quantities like P i ∆P j .We approach this question by considering averages of the form ⟨g(Φ ′ ) − g(Φ)⟩ in lowest order ∆t, with an arbitrary (smooth) function g of the system state.In analogy to (10), the probability of observing a certain value g(Φ ′ ) is given as such that we obtain Using our insights gained above, in Section 3.2, we expand this expression to lowest order in ∆t, where we use (50) in the last step.With the standard rules of ("non-stochastic") calculus, the expansion in lowest order would result only in the first two terms.The third term corresponds exactly to the additional contribution characteristic for the so-called Stratonovich rule of stochastic calculus [3,4].Denoting this rule by •, we can therefore write As a concrete example, we consider g(Φ) = P 2 :

Connection to the Fokker-Planck Equation
In Section 3.1, we prove that the effective system dynamics are Markovian on ∆t time scales; see Equation (12).Therefore, the dynamics also obey the Chapman-Kolmogorov equation [2,3,5], which, by the standard Kramers-Moyal expansion [2,3,5], is recast into the (forward) Kolmogorov equation (adjusted to our notation here), with the coefficients (using t ′ − t = ∆t as before) Here, p(Φ ′ , t ′ |Φ, t) is the transition probability density from Equation (10) (including explicitly the time arguments, but skipping the subscript E).The coefficients (62) are given by exactly the moments we already calculated, cf.Equation (18).From the results listed in Equation (53) (and by remembering that all other moments are O(∆t 2 )), we immediately obtain where we also used dϕ = d φ.Since these integrals on both sides are over all possible initial conditions of the bath and thus over all possible solutions ϕ t , they are identical, implying that A completely analogous argument, based on the combination of three sign changes under the mirror transformation in the integral on the right-hand side of (A1b), leads to These results are valid component-wise and at any point t in time along the trajectories ϕ t = (p t , q t ), in particular also at the initial time, i.e., for ϕ t = ϕ.
We finally point out that we cannot perform a corresponding calculation involving the solutions Φ t = (P t , Q t ) for the system evolution rather than the fixed (initial) value We consider an arbitrary function F(ϕ, Φ) of the phase-space variables (ϕ, Φ) and want to connect the integral dϕ δ ′ E − H bath (ϕ, Q) F(ϕ, Φ) to the microcanonical average of F(ϕ, Φ) over ρ mc (ϕ) (see Equation ( 8)).We achieve this by introducing an auxiliary integration and differentiation over the energy variable E so that we can perform a partial integration: 1 where, in the last step, we use the definition (9) of the temperature.For non-extensive observables F(ϕ, Φ) whose average does not depend on the size (i.e., energy) of the bath, the first term in the last line vanishes.This is in particular the case for "local" observables like short-ranged interaction forces between the system and bath.For such observables, we therefore find that 1 Ω(E) We remark that this relation is key for establishing the fluctuation-dissipation relation between thermal fluctuations and friction affecting the system particle.

Appendix C. The Different Representations for γ ij
Here, we establish that the two averages 1 k B T ∂H int (q,Q) ∂Q j ∂H int (q t ′ ,Q) ∂Q i mc and ∂ 2 H int (q t ′ ,Q) ∂Q j ∂Q i mc are identical (with corrections of O(∆t) or higher) for all t ≤ t ′ ≤ t + ∆t.We start our calculation from 1 k B T ∂H int (q,Q) ∂Q j ∂H int (q t ′ ,Q) ∂Q i mc by adding a zero term using (A7a): In the last step, we used the result (A9) from the previous Appendix.We now recall that H sys (Φ, λ) (see Equation ( 6)), such that Hence, we can rewrite the above result (A10) in terms of a spatial derivative of the δ function, and then continue with a partial integration in the Q j variable: The interaction Hamiltonian H int (q t ′ , Q) depends on Q j also via the initial point of the solution q t ′ = q t ′ (ϕ, Φ), such that there is an additional term ∑ µ,k ∂ 2 H int (q t ′ ,Q) ∂q µ,k ∂Q i ∂(q µ,k ) t ′ ∂Q j . However, this term is of order ∆t (at least), which follows from , and is absorbed into O(∆t).The boundary term (first line) vanishes, using (A7a).We are thus left with concluding the proof.As said already, the relation (written in lowest order ∆t) is valid for all t ≤ t ′ ≤ t + ∆t, in particular also for t ′ = t: