Conditional Fluctuation Theorem in Nonequilibrium Physics

• Conditional Fluctuation Theorem is a class of equalities that condition on system trajectories to produce refined bounds on entropy and free-energy in nonequilibrium systems.
• These theorems generalize classic results like Crooks and Jarzynski by integrating marginal observables, microstate constraints, and quantum corrections in their formulation.
• They provide practical methodologies for free-energy inference, variance reduction, and experimental design in both classical and quantum thermodynamics.

Conditional fluctuation theorems constitute a broad class of exact equalities and inequalities in nonequilibrium statistical mechanics and quantum thermodynamics. They generalize conventional fluctuation theorems by conditioning on subsets of system trajectories, marginal observables, microstate constraints, or information-theoretic structure, subsuming classic results such as the Crooks detailed theorem and Jarzynski equality as limiting cases. Through explicit conditioning, these theorems yield stronger second-law-type bounds, improved inference of free-energy differences, and new insights into the interplay between entropy production, information transfer, and irreversibility, both in classical and quantum settings.

1. Formal Structure of Conditional Fluctuation Theorems

Conditional fluctuation theorems arise when the ensemble of possible system histories (trajectories) is partitioned or coarse-grained according to a measurable criterion. Let $\Omega$ denote the space of all trajectories $\gamma$ under a protocol $\lambda(t)$ in contact with a thermal environment. For a subset $C \subseteq \Omega$, one defines the forward and reverse path-probability densities restricted to $C$, denoted $P[\,\gamma\,|\,C\,]$ and $R[\,\gamma\,|\,C\,]$ (Wimsatt et al., 2022). The total entropy-change functional over a trajectory, $\Sigma[\gamma]$, generally admits the path-based detailed fluctuation symmetry

$$\Sigma[\gamma] = \ln\left( \frac{P[\gamma]}{R[\gamma]} \right)$$

with underlying microscopic reversibility $$p[\gamma|z(0)] / r[\tilde\gamma|z(\tau)] = \exp(-\beta Q[\gamma])$$ for thermal systems.

Conditional theorems express identities or inequalities for the statistics of $\Sigma[\gamma]$ within $C$, typically in two main forms:

• Nominal Class Fluctuation Theorem (NCFT):

$$\langle \Sigma \rangle_C = \ln \frac{P(C)}{R(C)} + \mathcal{K}_C$$

where $$\mathcal{K}_C = \int_C dP[\gamma|C] \ln \frac{P[\gamma|C]}{R[\gamma|C]} \ge 0$$ is the class irreversibility.

• Exponential Class Fluctuation Theorem (ECFT):

$\langle e^{-\Sigma} \rangle_C = \frac{R(C)}{P(C)}$

These generalize single-trajectory (Crooks) and ensemble-averaged (Jarzynski) results, interpolating between them as $C$ varies from a singleton to the full path space (Wimsatt et al., 2022).

2. Generalization and Unification of Fluctuation Relations

Conditional fluctuation theorems unify a diverse array of classical and quantum results. They interpolate between detailed fluctuation theorems (DFT) for individual trajectories (e.g., Crooks) and integral theorems for entire ensembles (e.g., Jarzynski equality), and further extend to:

• Marginal and modified fluctuation theorems for observables not strictly satisfying the conditions for exact DFTs, leading to additional conditional averaging factors (Lahiri et al., 2014).
• Hierarchies of conditional FTs on outcome-resolved statistics in quantum-state tomography, capturing off-diagonal coherences and out-of-time-ordered fluctuations (Tsuji et al., 2018).
• Quantum conditional FTs formulated in terms of positive operator-valued measures (POVMs), energy reservoirs, and reference/demon systems, thus encompassing feedback, coherence, and information-theoretic structure (Aberg, 2016, Zhang et al., 2021, Yada et al., 2021).

A typical structure, as in the trajectory class fluctuation theorem (TCFT), is shown below for conditioning on a trajectory class $C$:

Conditioning	Fluctuation theorem type	Key identity
Single trajectory $\{\gamma\}$	Crooks DFT	$\ln[P(\gamma)/R(\gamma)] = \Sigma[\gamma]$
All trajectories $\Omega$	Jarzynski / Seifert IFT	$\langle e^{-\Sigma} \rangle = 1$, $\langle\Sigma\rangle = D_{KL}(P\Vert R) \ge 0$
Subset $C \subseteq \Omega$	Conditional / class FT	$\langle\Sigma\rangle_C \ge \ln[P(C)/R(C)]$, $\langle e^{-\Sigma}\rangle_C = R(C)/P(C)$

These relations subsume and sharpen standard second-law-like bounds and thermodynamic inequalities (Wimsatt et al., 2022, Lahiri et al., 2014).

3. Methodologies for Derivation and Implementation

The derivation of conditional fluctuation theorems proceeds from fundamental principles of microreversibility or quantum unitarity. Key technical steps include:

• Definition of forward and reverse path probabilities, conditional on specified initial and final distributions and possibly on measurement outcomes or path features.
• Partitioning trajectory space via measurable classes $C$, microstate labels, or functional observables.
• Application of Jensen's inequality and Kullback-Leibler divergence to obtain strengthened second-law forms.

In quantum settings (e.g., (Zhang et al., 2021, Aberg, 2016)), theorems are established at the level of stochastic process realizations following two-point measurement schemes, or at the operator level using completely positive maps on the reservoir, including post-selection and measurement back-action. Incorporation of continuous measurement and feedback employs stochastic master equation unravellings and introduces information-theoretic corrections, such as QC-transfer entropy (Yada et al., 2021).

4. Information-Theoretic and Marginal Conditioning

Conditional fluctuation theorems naturally extend to scenarios involving conditional statistics of correlated subsystems or information-exchange protocols:

• Conditioning on final microstates or correlated endpoints yields refined equalities such as:

$$\langle e^{-[\sigma + I_0]} \rangle_{x_\tau, y_\tau} = e^{-I_\tau(x_\tau, y_\tau)}$$

where $I_t(x_t, y_t)$ is the point-wise mutual information, quantifying information-theoretic correlations as part of the entropy budget (Jinwoo, 2019).

• In mesoscopic quantum transport or coupled currents, marginal fluctuation theorems for one channel conditioned on the statistics of another yield effective affinities and modified fluctuation symmetries, governed by timescale separation and interaction strength (Cuetara, 2013).
• In systems with an external reference (demon or memory), additional terms quantify the irreversibility associated strictly with maintaining or dissipating correlations, formalized as dissipative information and obeying their own fluctuation theorems (Zhang et al., 2021).

These constructions reveal how conditional thermodynamic and information-theoretic costs are allocated across subsystems, measurement outcomes, and experimental coarse-grainings.

5. Applications: Free-Energy Inference, Variance Reduction, and Experimental Design

A principal practical outcome of conditional fluctuation theorems is improved estimation of free-energy differences and other thermodynamic quantities from finite data, especially in the presence of rare events that dominate standard exponential averages. Conditional estimators exploit restricted classes $C$ of high measure to suppress variance:

$$\Delta F_C = -\frac{1}{\beta} \ln \langle e^{-\beta W} \rangle_C + \frac{1}{\beta} \ln \frac{R(C)}{P(C)}$$

This approach, as demonstrated in flux qubit information engine experiments, yields unbiased free-energy estimates with orders-of-magnitude lower variance compared to full-ensemble Jarzynski estimators, thus mitigating sampling difficulties (Wimsatt et al., 2022).

Conditional theorems also guide:

• Optimal selection of trajectory classes for tight second-law bounds (avoidance of rare events and narrow fluctuations within $C$),
• Quantitative characterization and separation of thermodynamic and informational entropy production in quantum feedback and Maxwell's demon scenarios (Zhang et al., 2021, Yada et al., 2021),
• Diagnosis of information flow or back-action in quantum transport and measurement-driven quantum systems (Cuetara, 2013, Yada et al., 2021).

6. Quantum Generalizations and Coherence-Resolved Relations

In quantum settings, conditional fluctuation theorems apply to both diagonal and off-diagonal (coherence) statistics, incorporating measurement back-action, time-reversal antiunitaries, reference systems, and conditional mutual information structure:

• State-resolved and process tensor approaches yield infinite hierarchies of fluctuation theorems, connecting to out-of-time-ordered correlators and quantum chaos diagnostics (Tsuji et al., 2018).
• Conditional (class, feedback, or measurement-outcome) quantum Crooks and Jarzynski relations are available using the Gibbs map formalism, POVMs, and post-selection protocols (Aberg, 2016).
• Generalizations involving continuous measurement and real-time feedback control are governed by extended fluctuation theorems with explicit QC-transfer entropy corrections (Yada et al., 2021).

These quantum FTs systematically reproduce classical results in suitable limits, reveal uniquely quantum signatures via coherence-resolved fluctuation statistics, and provide theoretical support for current circuit-QED and quantum information experiments (Zhang et al., 2021, Yada et al., 2021).

7. Modified and Feedback-Corrected Fluctuation Theorems

When only marginal or partial trajectory statistics are available, or when feedback is imposed based on intermediate measurements, one frequently obtains modified detailed fluctuation theorems. These involve additional conditional expectation factors—distinct from "exact" fluctuation theorems—typically of the form: $$\frac{P_f(A)}{P_r(-A)} = \exp(A) \left\langle e^{-B} \right\rangle_{P_f(B|A)}$$ where $A+B$ is the total entropy production, and $B$ the unobserved complement (Lahiri et al., 2014). In the presence of measurement and feedback, stochastic mutual information terms enter, yielding: $$\langle e^{-A-I} \rangle_f = \langle e^{-B} \rangle_r$$ with $I$ representing the information extracted by the feedback protocol. Only for total entropy or work, where $A$ completely spans the total entropy production, does the extra factor reduce to unity, thus recovering standard Crooks or Seifert forms.

• "Trajectory Class Fluctuation Theorem" (Wimsatt et al., 2022)
• "Fluctuation theorem for quantum electron transport in mesoscopic circuits" (Cuetara, 2013)
• "Fluctuation Theorem of Information Exchange within an Ensemble of Paths Conditioned on Correlated-Microstates" (Jinwoo, 2019)
• "Fluctuation theorem for quantum-state statistics" (Tsuji et al., 2018)
• "Conditional entropy production and quantum fluctuation theorem of dissipative information: Theory and experiments" (Zhang et al., 2021)
• "Fully quantum fluctuation theorems" (Aberg, 2016)
• "Quantum Fluctuation Theorem under Continuous Measurement and Feedback" (Yada et al., 2021)
• "Derivation of the not-so-common fluctuation theorems" (Lahiri et al., 2014)