Thermodynamics of Active Systems

• Active systems are collections of self-propelling particles that continuously consume energy, leading to non-equilibrium behaviors and ambiguous thermodynamic definitions.
• Different temperature definitions (velocity-based vs. position/dynamics-based) highlight distinct aspects of kinetic energy and structural fluctuations, especially far from equilibrium.
• Mass rescaling and careful experimental design allow for an operational collapse of temperature measures, enabling effective modeling and mapping to equilibrium analogs.

Active systems are collections of particles or agents that autonomously consume energy to generate motion or internal degrees of freedom. This continual energy consumption drives such systems far from equilibrium, resulting in persistent dissipation, emergent structures, and collective behaviors that are not captured by classical equilibrium thermodynamics. In the context of thermodynamics, describing active systems poses fundamental challenges: conventional definitions of quantities such as temperature, entropy, and free energy may become ambiguous or even break down. Nevertheless, a growing body of research has systematically investigated how thermodynamic concepts can be rigorously and usefully generalized for active systems, establishing principles, formalisms, and experimental signatures that characterize their energetic and statistical properties.

1. Definitions of Temperature in Active Systems

Unlike in equilibrium thermodynamics, where all operational thermometers coincide, active systems can support multiple, incompatible definitions of temperature depending on which degrees of freedom or measurements are considered. The paper "How to define temperature in active systems?" (Hecht et al., 27 Jul 2024) surveys three broad families of definitions:

• Velocity-based temperatures: The kinetic temperature

$$T_{\rm kin} = \frac{1}{Nd} \sum_{i=1}^N m \langle (\mathbf{v}_i - \bar{\mathbf{v}})^2 \rangle$$

and, as a variant, the fourth-moment kinetic temperature

$$T_{\rm kin4} = \frac{1}{2}m\sqrt{\frac{4}{Nd(d+2)} \sum_{i=1}^N \langle (\mathbf{v}_i-\bar{\mathbf{v}})^4 \rangle}$$

are based on measuring single-particle velocity fluctuations. There is also the configurational temperature, defined in terms of the potential energy landscape,

$$T_{\rm conf} = \langle |\nabla U_{\rm tot}|^2 \rangle / \langle \nabla^2 U_{\rm tot} \rangle.$$

• Position-based and dynamic temperatures: The virial temperature

$$T_{\rm vir} = \frac{2}{Nd} \langle \mathcal{V} \rangle$$

exploits the virial theorem, while the oscillator temperature for confined particles is given by

$T_{\rm osc} = k \langle r^2 \rangle$

(with $k$ the harmonic potential strength). The Einstein temperature,

$T_{\rm Ein} = \gamma_t D_{\rm eff}$

is based on the effective long-time diffusion coefficient, and the effective temperature is defined through fluctuation-dissipation relations as

$$T_{\rm eff} = \lim_{t \gg 1} \frac{\langle \text{MSD}(t) \rangle}{2d \, \chi(t)}$$

where $\chi(t)$ is the integrated susceptibility.

Simulations demonstrate that, near equilibrium, all notions coincide and yield the bath temperature $T_b$. Far from equilibrium, these definitions generally produce different values, categorizing into two classes: (i) velocity-based (kinetic, fourth-moment, configurational), and (ii) position/dynamics-based (virial, oscillator, Einstein, effective). The two classes are strongly separated when activity is high or when the particle mass is small, but approximately collapse onto each other upon suitable mass rescaling, even far from equilibrium.

2. Thermodynamic Interpretation and Relevance

These distinct temperature definitions correspond to fundamentally different aspects of non-equilibrium fluctuations in active systems. The velocity-based class measures kinetic energy storage and its fluctuations—quantities that are heavily mass-dependent in active Brownian dynamics and, for non-interacting active particles, scale as $m v_0^2 / 2$, with $v_0$ the propulsion speed. Conversely, position-based or response-based definitions probe the long-time diffusive or configurational behavior, are less sensitive to the mass, and depend more on the interplay between active driving, environmental dissipation, and confining potentials.

This dichotomy is not merely academic: velocity-based and position/dynamics-based temperatures can diverge significantly at high Péclet number (Pe) or for light particles, and only coincide with the bath temperature in the $Pe \to 0$ or large-mass limits. Effective thermodynamic descriptions of active phase behavior (such as mapping to a passive system at an effective temperature or measuring heat capacities) must be carefully constructed, choosing the relevant definition that best describes the process or observable of interest (Hecht et al., 27 Jul 2024).

3. Classes and Collapse of Temperatures

Quantitative comparisons across regimes reveal that, aside from special limits, temperatures cluster into two robust classes:

Class	Examples	Primary Sensitivity
Velocity-based	$T_{\rm kin}$, $T_{\rm kin4}$, $T_{\rm conf}$	Strongly depends on $m$, activity
Position/dynamics-based	$T_{\rm vir}$, $T_{\rm osc}$, $T_{\rm Ein}$, $T_{\rm eff}$	Weakly depends on $m$; sensitive to interactions and confining forces

After dividing the velocity-based temperatures by the particle mass, both classes quantitatively align over a wide range of Péclet number and particle mass, except when Pe is extremely large or the system is close to equilibrium. This collapse suggests an operational self-consistency within each class and offers a route to compare measurements from different experiments or simulations.

4. Implications, Measurement, and Applicability

The ambiguity in temperature definitions carries implications for the measurement and control of active systems:

• Experimental measurements: In practice, choice of thermometer (velocity- or position-based) can strongly affect the measured "temperature." For example, velocity-based methods are easily implemented in high-speed tracking but can be sensitive to mass and non-Gaussian distributions, while position-based methods using tracers in confining potentials are common in micromanipulation and calorimetry.
• Modeling and simulation: For mapping active matter to effective equilibrium models (as in active heat engine studies or phase separation analysis), the temperature definition should match the relevant steady-state observable; for example, using $T_{\rm osc}$ in confined systems or $T_{\rm Ein}$ for large-scale diffusion.
• Theory and design: The existence of two robust classes of temperature highlights the need for careful selection of the appropriate definition for a given non-equilibrium process. For energy transfer or phase transitions involving kinetic modes, velocity-based temperatures are relevant; for transport, structural transitions, or long-time dynamics, position/dynamics-based definitions are likely more appropriate.
• No universal temperature: The lack of a unique temperature in active matter underscores its fundamental departure from equilibrium thermodynamics. Attempts to assign a single $T$ to all aspects of the system are generally unfounded except in the equilibrium limit; rather, the choice must be guided by the physical context and the class of observable probed.

5. Advantages and Disadvantages of Definitions

Each class of temperature definition comes with particular operational strengths and pitfalls:

Definition Class	Advantages	Disadvantages
Velocity-based	Direct access via single-particle velocities; analytical tractability for low-density and non-interacting limit	Strong mass dependence, ambiguity in interacting or non-Gaussian cases
Position/dynamics-based	Mass-insensitive, robust to confining or interaction effects, compatible with imaging methods	May require tracer-specific calibration; confinement parameter sensitivity; $T_{\rm eff}$ computationally expensive (needs response functions)

For high-activity or strongly driven systems, experiments should be designed to minimize sensitivity to the specific weaknesses of the selected definition. For example, adjusting tracer size and confining strength in oscillator-based approaches, or using collective position fluctuations to extract effective temperatures in large systems.

6. Practical Implications and Applications

The paper and classification of temperature definitions yield direct benefits:

• For experimental design, knowing which observable matches the intended "thermometer" guides sample preparation, measurement protocols, and interpretation.
• In active engine settings, controlling heat capacity or energy flux depends sensitively on the operational temperature relevant to the energy transduction mechanism.
• For theoretical models seeking coarse-grained, field-theoretic, or stochastic thermodynamic descriptions, the explicit recognition that two classes exist—each capturing different aspects of activity-induced fluctuations—facilitates construction of hydrodynamic and thermodynamic theories grounded in measurable quantities.

Ultimately, this comprehensive delineation of temperature reflects a maturing statistical mechanics of active matter: although classical thermodynamic equivalence is lost, a careful selection and classification of operational definitions enable quantitative, experimentally relevant, and self-consistent energetic interpretations for a wide range of non-equilibrium active systems (Hecht et al., 27 Jul 2024).