Mutual-information and nonperturbative interpretation of the entropy-production balance in the stochastic thermodynamics of fractional Brownian motion

An overdamped particle driven by fractional Gaussian noise (fGn) is the paradigmatic non-Markovian Gaussian dynamics exhibiting anomalous diffusion, and its noise autocorrelation is unrelated to its friction, so no fluctuation-dissipation relation of the second kind holds and no unique temperature exists. Khadem, Klages and Klapp [Phys. Rev. Research 4, 043186 (2022)] showed that the heat dissipated into a constant-temperature medium then differs from the log-ratio of forward and backward path probabilities, and that consistency of the integral fluctuation relation is restored only by introducing a time-nonlocal temperature kernel (the functional inverse of the fGn covariance) and a generalized heat functional, evaluated so far only to first order in the deviation of the Hurst exponent H from the Brownian value 1/2. The entropy-production balance then contains an additional entropy-like term, and the authors pose as an explicitly open question whether it can be expressed quantitatively as a mutual-information production between particle and bath dynamics. No published work (2022-2026) gives this term an information-theoretic interpretation, constructs the generalized temperature and heat beyond first order, or refutes the possibility. The problem is open.

Background

Stochastic thermodynamics (ST) assigns heat, work and entropy production to individual trajectories of Langevin-type dynamics and yields exact constraints such as the integral fluctuation relation (IFR) $\langle e^{-\Delta S^{tot}}\rangle=1$ for the total entropy production along a trajectory (Seifert, Rep. Prog. Phys. 75, 126001 (2012)). For a standard Brownian particle with mobility $\mu$ and white noise of strength $D$ , the fluctuation-dissipation relation of the second kind (FDR2) — noise autocorrelation proportional to the (delta-like) friction kernel — defines the medium temperature $T_0=D/\mu$ and allows medium entropy production to be computed as dissipated heat divided by $T_0$ . Fractional Brownian motion (FBM) is the Gaussian process with stationary increments and autocorrelation proportional to $t_1^{2H}+t_2^{2H}-|t_1-t_2|^{2H}$ , where the Hurst exponent $H$ gives subdiffusion for $H<1/2$ and superdiffusion for $H>1/2$ ; its formal derivative is fractional Gaussian noise (fGn), whose autocorrelation is nonlocal in time (proportional to $|t_1-t_2|^{2H-2}$ ). FBM and fGn-driven motion are widely observed in crowded fluids and biological systems. In the system considered by Khadem, Klages and Klapp, an overdamped particle obeys $\dot{x}(t)=\mu F(x(t),\lambda(t))+\xi^H_{fGn}(t)$ with an external protocol $\lambda(t)$ : the fGn is prescribed exogenous noise, so FDR2 is broken and the system is 'athermal' in the precise sense that temperature and noise strength are unrelated. (As a warm-up they first treat the Markovian case of time-dependent diffusivity, e.g. scaled Brownian motion, where a time-dependent temperature $T(t)=D(t)/\mu$ keeps the standard IFR intact.) For the fGn-driven particle the log-ratio of forward and backward path probabilities equals a double time integral $\int dt_1\int dt_2\,F(t_1)\,T^{-1}(t_1-t_2)\dot{x}(t_2)$ , which can be read as a medium entropy production involving the 'generalized temperature' kernel $T^{-1}(t_1-t_2):=2\mu\langle\xi^H_{fGn}(t_1)\xi^H_{fGn}(t_2)\rangle^{-1}$ — the functional inverse of the fGn covariance, in a relation the authors call a generalized FDR2. This route, however, disagrees with the conventional dissipation $\Delta Q=\int_0^t dt'\,F\,\dot{x}$ at constant $T_0$ ; the IFR survives only in the modified form $\langle e^{-\Delta S^{tot}+\Delta S^I}\rangle=1$ with the additional entropy-like term $\Delta S^I_{[x]}=\mathbb{Q}-\Delta Q$ , where $\mathbb{Q}$ is the generalized heat functional built from the kernel. Khadem et al. evaluate $T$ and $\mathbb{Q}$ by perturbation expansion in $\epsilon=H-1/2$ , finding $\mathbb{Q}(t)=\Delta Q(t)-\epsilon({}_{\dot{x}}\Sigma_{F}+{}_{F}\Sigma_{\dot{x}})+O(\epsilon^2)$ , with the first-order corrections interpretable as extra heat exchanges through a retarded force and a retarded velocity. For analogous Gaussian active-bath systems, Dabelow, Bo and Eichhorn [Phys. Rev. X 9, 021009 (2019)] interpreted the additional entropy-like terms as a mutual-information production between particle and bath degrees of freedom (the time-differentiated mutual information between the two sectors), which is well defined there because the bath has explicit dynamics coupled to the particle. Whether the same interpretation transfers to the fGn-driven particle is the source's stated open question, and it faces real obstacles: the fGn is exogenous with long-range power-law correlations, so there is no canonical bath dynamics (no finite-dimensional Markovian embedding, i.e. no representation of the memory as ordinary Markovian dynamics of finitely many auxiliary variables, reproduces it exactly); FBM is not a semimartingale (the class of processes on which the standard Itô calculus of stochastic integrals and measure changes is built), so path probabilities involve fractional integrals with delicate Jacobian and boundary terms; the covariance inverse degenerates to distributions at $H=1/2$ ; and for $H>1/2$ the generalized heat functional requires additional boundary conditions involving fractional derivatives at $t=0$ , with the perturbation theory relying on a short-time cut-off $\tau$ that the authors fix at first order by a particular choice ( $\tau=e^{-\zeta}$ ) but whose treatment beyond first order remains to be justified. Related fluctuation relations for Gaussian dynamics satisfying FDR2 were established earlier by Chechkin, Lenz and Klages [J. Stat. Mech. (2012) L11001].

Problem Statement

Setting (from the source paper): an overdamped point particle with mobility $\mu$ obeys $\dot{x}(t)=\mu F(x(t),\lambda(t))+\xi^H_{fGn}(t)$ under an external protocol $\lambda(t)$ , where the fractional Gaussian noise $\xi^H_{fGn}$ has autocorrelation $2K_H(2H-1)|t_1-t_2|^{2H-2}+4K_HH|t_1-t_2|^{2H-1}\delta(t_1-t_2)$ with Hurst exponent $0<H<1$ and noise strength $K_H>0$ ; no FDR2 links the noise to the friction. Two inequivalent definitions of medium entropy production then coexist: the conventional dissipation $\Delta Q=\int_0^t dt'\,F(x(t'),\lambda(t'))\dot{x}(t')$ at a constant temperature, and the path-probability route involving the time-nonlocal temperature kernel $T^{-1}(t_1-t_2):=2\mu\langle\xi^H_{fGn}(t_1)\xi^H_{fGn}(t_2)\rangle^{-1}$ and the generalized heat functional $\mathbb{Q}(t)$ , which restores the integral fluctuation relation in the form $\langle e^{-\Delta S^{tot}+\Delta S^I}\rangle=1$ with the additional term $\Delta S^I_{[x]}=\mathbb{Q}-\Delta Q$ , currently evaluated only to first order in $\epsilon=H-1/2$ .

The problem (the source's stated open question): information-theoretic interpretation of the additional term. Determine whether $\Delta S^I_{[x]}$ (in finite-time or rate form) admits an exact quantitative expression as a mutual-information production between the particle trajectory and a bath or noise-generating process that reproduces the fGn autocorrelation. A constructive answer must build the joint process and the information functional explicitly, with the Hurst range, force/protocol class, time horizon and initial conditions stated. The only admissible negative answer is a proof that no such representation exists for any joint construction whose bath or noise-generating process reproduces the fGn autocorrelation; restricting the negation to a narrower, solver-chosen class does not resolve the question — in particular, proving that no finite-dimensional Markovian embedding reproduces the fGn exactly (a fact already recorded in the Background and in the difficulties below) leaves the source's question untouched, since mutual-information productions of the Dabelow-Bo-Eichhorn type may involve infinite-dimensional or age-structured bath dynamics.

Acceptance criteria for any answer: it must reduce at $H=1/2$ to standard Brownian stochastic thermodynamics (the information term vanishing and $\mathbb{Q}\to\Delta Q$ ); reproduce at $O(\epsilon)$ the published first-order expressions, in particular $\mathbb{Q}(t)=\Delta Q(t)-\epsilon({}_{\dot{x}}\Sigma_{F}+{}_{F}\Sigma_{\dot{x}})+O(\epsilon^2)$ (where ${}_{\dot{x}}\Sigma_{F}$ and ${}_{F}\Sigma_{\dot{x}}$ are the retarded-force and retarded-velocity heat-exchange terms of the source paper); and leave the integral fluctuation relation intact. The nonperturbative construction of $T(t_1-t_2)$ and $\mathbb{Q}(t)$ beyond first order in $\epsilon$ — including how the fractional-derivative boundary conditions at $t=0$ required for $H>1/2$ are imposed, how the short-time cut-off $\tau$ is handled beyond first order, and any operational meaning (an in-principle measurement or inference protocol from trajectories) of these memory-bearing quantities — is an adjacent open direction indicated by the source's limitation of these quantities to zeroth and first order, not a required deliverable of this problem. The authors' separately proposed program of deriving nonlinear response relations for general non-Markovian systems from fluctuation relations is likewise an adjacent open direction, not a required deliverable of this problem.

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