Speed limits to fluctuation dynamics

Non-equilibrium statistical mechanics studies systems with randomness from stochastic noise or quantum uncertainty. Key characteristics are the mean and fluctuations of measured observables. Out-of-equilibrium, the observable distribution changes, impacting the mean and fluctuation's time evolution. Recent research has focused on rigorous bounds for the mean's rate of change—speed limits—building upon Mandelstam and Tamm's early work. These speed limits, often tighter than state-based constraints, provide experimentally relevant bounds. However, previous speed limits don't clearly predict observable fluctuations (e.g., standard deviation), which aren't simply state-independent means. Fluctuation growth is crucial for understanding non-equilibrium dynamics; for instance, particle spreading involves both mean displacement and position fluctuation dynamics, with mean-based bounds being loose. Fluctuation dynamics also relate to entanglement and coherence in quantum systems, aspects not captured by the mean. This paper addresses the open question of elucidating the dynamical behavior of observable fluctuations by developing a theory for rigorous bounds on fluctuation dynamics.

The study of speed limits in non-equilibrium systems has a rich history, with early contributions focusing on the limitations of the rate of change of the mean value of an observable. Mandelstam and Tamm's work laid the foundation for these investigations, which have continued to evolve with recent research refining bounds using concepts like standard deviation and Fisher information. These bounds are often more experimentally relevant than those based solely on the system's state. However, a significant gap existed in understanding the dynamics of fluctuations, specifically how the speed of fluctuation growth is constrained. This paper bridges this gap by developing a novel theoretical framework that provides rigorous bounds on the dynamics of fluctuations.

The paper establishes a theoretical framework for bounding the dynamics of observable fluctuations. A key concept is the introduction of a "velocity observable," denoted as *V_A*, chosen such that its mean equals the time derivative of the mean of the observable *A*. The core result is an inequality showing that the speed of the standard deviation of *A* (dσ_A/dt) is bounded by the standard deviation of *V_A* (σ_{V_A}), expressed as dσ_A/dt ≤ cσ_{V_A}, where *c* is a constant (1 or 1/2 depending on the system). This inequality implies a tradeoff between the speeds of the mean and standard deviation: they cannot both be simultaneously fast. This tradeoff is concisely represented as (d⟨A⟩/dt)² + (dσ_A/dt)² ≤ *v_A²σ_A²* = *C_A*. The paper provides a general proof of these inequalities, identifying sufficient conditions for their validity by focusing on how to appropriately choose the velocity observable. The approach uses inner products to represent expectation values of observables, considering continuous classical systems, classical discrete systems, and quantum systems. The time evolution of probability is described by dp/dt = L[p], with a dual map C satisfying (A|L[p]) = (C[A]|p). The velocity observable is then defined as V_A = Å + C[A]. The paper provides detailed examples of its application to various physical systems including irreversible thermodynamics, classical and quantum hydrodynamics, unitary quantum dynamics, and nonlinear population dynamics. For higher-order moments, a similar inequality is derived, and connections to Wasserstein geometry are explored. The methodology extends to discrete systems using a discrete version of the local conservation law of probability. The framework is applied to quantum transport in many-body systems, establishing tighter bounds than previous approaches. For time-dependent observables, the theory provides speed limits on the amount of work done in quantum dynamics. The paper also examines entanglement growth in non-interacting fermions, relating it to number fluctuations at the boundary of a subsystem and offering tighter bounds than prior methods. The paper explores limits stemming from classical information theory, using the classical Fisher information and showing the time-information uncertainty relation for mean and fluctuation. Finally, the theory is extended to general quantum dynamics, including dissipation, although a suitable velocity observable for this case remains elusive.

The paper's central finding is the establishment of a new principle governing the speed of fluctuation dynamics in non-equilibrium systems. This principle states that the speed at which the standard deviation of an observable changes is always less than or equal to the standard deviation of a suitably defined 'velocity observable.' This result is expressed mathematically through inequalities that take different forms depending on the specific system under consideration. For example, for certain dynamics, the speed of the standard deviation of an observable *A* is bounded by the standard deviation of an appropriately chosen velocity observable *V_A*: dσ_A/dt ≤ cσ_{V_A}. The constant *c* is 1 or 1/2, depending on the details of the system. Furthermore, the paper discovers a trade-off relationship between the speeds of the mean and standard deviation, indicating that both quantities cannot simultaneously change rapidly. This relationship is mathematically expressed as (d⟨A⟩/dt)² + (dσ_A/dt)² ≤ *C_A*, where *C_A* is a quantity that depends on the system and the observable *A*. The paper demonstrates that *C_A* can be further bounded by other physically relevant quantities, such as the kinetic energy in hydrodynamics and the entropy production rate in thermodynamics. The paper illustrates the application of these results in various physical scenarios: 1. **Irreversible Thermodynamics:** The theory is applied to Brownian particle systems, showing that rapid changes in the mean and fluctuation require significant irreversibility. A comparison with previous speed limits reveals that the new inequalities provide significantly tighter bounds, particularly when the changes in the standard deviation are comparable to or larger than those in the mean. 2. **Classical and Quantum Hydrodynamics:** The results apply to both classical and quantum hydrodynamics. The bound *C_A* is related to the kinetic energy, indicating that small kinetic energy implies slow changes in both mean and fluctuation. 3. **Quantum Work Extraction:** The inequalities provide speed limits for the amount of work done in time-dependent unitary quantum dynamics. This implies that the work and fluctuation change are upper-bounded given the control cost of changing the Hamiltonian. 4. **Entanglement Growth:** The theory provides useful bounds for the growth of entanglement entropy in non-interacting fermions. The bounds are expressed in terms of number fluctuations at the boundary of a subsystem, offering a practically advantageous method for evaluating entanglement since it doesn't necessitate evaluating quantities across the entire bulk region. This leads to tighter bounds compared to previous, state-independent approaches. 5. **Nonlinear Population Dynamics:** The theory is extended to nonlinear population dynamics, where the fluctuation relations are distinct from existing Price equations. A specific case of evolutionary dynamics without mutation reveals a connection between the variance of the growth rate and the changes in higher-order moments. 6. **General Quantum Dynamics:** While a universal velocity observable is not found for general quantum dynamics (which includes dissipation), an inequality relating the speeds of the mean and standard deviation to the quantum Fisher information is derived.

The findings of this paper significantly advance the understanding of fluctuation dynamics in non-equilibrium systems. The established inequalities provide more precise and experimentally relevant constraints on fluctuation growth compared to prior approaches focusing solely on the mean. The discovery of the tradeoff relation between the speeds of the mean and standard deviation offers a new perspective on non-equilibrium dynamics, suggesting a fundamental limitation on how quickly both quantities can change simultaneously. The broad applicability of the theory, demonstrated across various physical systems from classical to quantum and from simple to many-body systems, underlines its fundamental nature. The connection to information-theoretic quantities like the Fisher information further strengthens the theoretical foundation and offers potential connections to the field of quantum metrology. The tighter bounds derived for entanglement growth and quantum transport highlight the practical utility of the theory, providing more efficient methods for estimating these quantities in experiments. The limitations of previous approaches to handle processes with non-negligible dispersion and those where the mean and fluctuation change at comparable rates are overcome by the proposed inequalities.

This paper presents a fundamental principle governing the speed of fluctuation dynamics in non-equilibrium systems. It establishes that the speed of fluctuation is bounded by the fluctuation of an appropriate velocity observable. This principle is expressed as a tradeoff relation between the speeds of the mean and standard deviation, bounded by a quantity further constrained by physical quantities (e.g., entropy production) or information-theoretic measures (Fisher information). Future work will focus on extending the theory to broader classes of dynamics, such as general open quantum systems and discrete-time dynamics, and exploring its application to controlling complex systems at the level of fluctuations.

While the theory offers a significant advancement, certain limitations exist. The theory's applicability to general open quantum systems is partially addressed, but a universal velocity observable remains elusive for this broad class of systems. The extension to discrete-time dynamics also represents an area for future research. Furthermore, while the paper focuses on standard deviation, the practical applicability of the results for higher-order moments might need further investigation.