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Equation of Motion for Open Thermodynamic Systems and Its Dynamical Equivalence with Mechanical and Electrical Systems

This paper derives an explicit equation of motion for open thermodynamic systems directly from the first law of thermodynamics, demonstrating their dynamical equivalence to driven damped harmonic oscillators and RLC circuits, and establishing a unified Lagrangian framework that reveals a common structural foundation across thermodynamic, mechanical, and electrical systems.

Original authors: E Aydiner

Published 2026-09-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: E Aydiner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the physical world, we often sort phenomena into neat categories: the swinging of a pendulum belongs to mechanics, the flow of electricity through a wire belongs to circuits, and the heating of a gas belongs to thermodynamics. For centuries, scientists have treated these as separate languages, each with its own rules for how energy moves and changes. Yet, deep within the laws that govern our universe, there is a persistent suspicion that these different realms might be speaking the same underlying dialect. The first law of thermodynamics, a cornerstone of physics, tells us how the internal energy of a system changes when heat is added or work is done. It is a powerful statement, but it is usually written as a static balance sheet, a snapshot of energy accounting that does not explicitly show how a system moves or evolves over time. While we have clear, dynamic equations that describe how a spring bounces or how a circuit responds to a voltage, a similar equation of motion for a thermodynamic system has remained elusive. This gap left a fundamental question unanswered: can the laws of heat and energy be rewritten to show the actual rhythm of change, revealing a hidden dynamical structure similar to the ones we see in mechanics and electricity?

A researcher at Istanbul University has now answered this question by showing that the first law of thermodynamics can indeed be transformed into a dynamic equation of motion for an open system—one that exchanges heat and particles with its surroundings. By carefully analyzing how a system responds when it is slightly disturbed from a state of balance, the author demonstrated that the complex interactions of entropy, pressure, and particle flow can be distilled into a single, unified mathematical form. They found that the way a thermodynamic system evolves is not a chaotic or unique process, but rather follows the exact same structural pattern as a mechanical oscillator, like a mass on a spring, and an electrical circuit containing a resistor, an inductor, and a capacitor. This discovery does not mean that heat is literally a spring or that temperature is a voltage; rather, it reveals that the fundamental organization of how these systems resist change, store energy, and dissipate energy is identical.

The researcher began by breaking down the first law of thermodynamics into its three main components: the change in entropy, the change in volume, and the exchange of particles. In a standard thermodynamic description, these are treated as separate contributions to the total energy. However, the researcher realized that if they looked at how these quantities behave when a system is near equilibrium, they could identify specific roles for each. The change in entropy, driven by the flow of heat, was found to act like inertia. Just as a heavy mass resists changes in its speed, the thermodynamic system resists changes in its flow of entropy, creating an inertial effect that depends on the rate of change. The exchange of particles between the system and its environment was identified as the source of dissipation. Much like friction slows down a moving object or electrical resistance slows down a current, the flow of particles generates a drag that dissipates energy. Finally, the pressure response of the system to changes in volume was shown to act as a restoring force. When the system is compressed or expanded, the pressure pushes back, trying to return the system to its original state, just as a spring pulls a mass back to its center.

By combining these three distinct thermodynamic behaviors, the researcher derived a single equation that governs the motion of the system. This equation describes a generalized coordinate, a single variable that tracks the state of the system, moving under the influence of an inertial term, a dissipative term, and a restoring term. The result is a perfect match for the equations that describe a damped harmonic oscillator in mechanics and a driven RLC circuit in electronics. The study further showed that this equivalence holds true even when the system is driven by an external source, such as a changing temperature or pressure, which acts like a driving force in the mechanical or electrical analogues. The researcher also constructed a Lagrangian formulation for the thermodynamic system, a mathematical framework usually reserved for mechanics and electromagnetism, which successfully described the system's energy in terms of kinetic-like and potential-like components. This confirmed that the thermodynamic system possesses a canonical structure identical to the other two, with defined momentum and position pairs that evolve in the same way.

The significance of this work lies in how it changes our understanding of the relationship between different branches of physics. Previously, the similarity between mechanical, electrical, and thermodynamic systems was often viewed as a convenient analogy, a way to use one field to solve problems in another. This paper argues that the similarity is not merely a coincidence of mathematics but a fundamental truth about how nature organizes energy. The author emphasizes that they did not start by assuming the thermodynamic system would look like a spring; instead, they started with the first law of thermodynamics and let the math reveal the structure. The resulting equation of motion emerged directly from the thermodynamic principles themselves, proving that the inertial, dissipative, and restoring behaviors are intrinsic properties of thermodynamic evolution near equilibrium.

This finding suggests that the first law of thermodynamics contains more dynamical information than is typically recognized. It implies that the laws governing heat and energy are not just about static balances but are capable of describing the actual trajectory of a system as it moves through time. The researcher notes that this result is specific to systems near equilibrium, where the responses are linear and predictable. They do not claim that this equation applies to every possible thermodynamic situation, particularly those far from equilibrium where chaos and nonlinearity might dominate. However, within the realm of near-equilibrium physics, the study establishes a rigorous bridge between thermodynamics and classical dynamics. It shows that despite their different physical origins—whether it is the vibration of a mass, the flow of electrons, or the exchange of heat and matter—these systems share a universal dynamical skeleton. The work provides a new perspective on the first law, revealing it as a generator of motion that can be expressed with the same clarity and power as the equations that describe the swinging of a pendulum.

Technical Summary: Equation of Motion for Open Thermodynamic Systems

Problem Statement
The first law of thermodynamics fundamentally describes changes in the internal energy of a system. However, its standard compact form ( $dU = TdS - pdV + \mu dN$ ) does not explicitly reveal the system's dynamics or provide a corresponding equation of motion. While explicit second-order equations of motion are well-established for mechanical systems (damped harmonic oscillators) and electrical systems (driven RLC circuits), an analogous dynamical formulation for thermodynamic systems has not been directly derived from the first law. This work addresses the question of whether a thermodynamic system exchanging heat and particles with its surroundings can be described by an explicit equation of motion that shares the mathematical structure of mechanical and electrical oscillators.

Methodology
The author employs a dimensional reduction approach to transform the first law of thermodynamics into a second-order dynamical equation. The derivation proceeds through the following steps:

Linear-Response Framework: The analysis is conducted within a near-equilibrium linear-response framework. The system is treated as an open thermodynamic system where temperature is evaluated at a reference equilibrium value ( $T \approx T_0$ ), while pressure deviations ( $\delta p$ ) and chemical potential differences ( $\Delta \mu$ ) are treated as dynamical quantities.
Generalized Coordinate: A generalized thermodynamic coordinate $\phi(t)$ is introduced to parametrize the effective thermodynamic mode. This coordinate is not identified with a single state variable (like $S$ , $V$ , or $N$ ) but serves as a common variable for the reduced description.
Decomposition of Energy Contributions: The rate of change of internal energy ( $\dot{U}$ $\dot{U}$ ) is decomposed into three distinct contributions, each analyzed separately to identify inertial, restoring, and dissipative terms:
- Entropy Contribution (Inertial Term): Utilizing Extended Irreversible Thermodynamics (EIT), entropy is expanded as a function of dissipative fluxes. The deviation of entropy from its equilibrium value is shown to be quadratic in the generalized flux ( $\dot{\phi}$ ). Differentiation yields a term proportional to $\dot{\phi}\ddot{\phi}$ , which is identified as an inertial energy-storage mechanism analogous to kinetic energy.
- Pressure-Volume Contribution (Restoring Term): The pressure response to volume displacement near a stable equilibrium is analyzed. Using the linear response of pressure to volume changes, a term proportional to $\phi\dot{\phi}$ is derived. This represents a restoring force analogous to the elastic potential energy of a spring.
- Particle-Transfer Contribution (Dissipative Term): Irreversible particle exchange between subsystems is modeled using Onsager's linear irreversible thermodynamics. The chemical potential difference is shown to be linearly proportional to the particle transfer rate ( $\dot{\phi}$ ), yielding a term proportional to $\dot{\phi}^2$ . This represents irreversible dissipation analogous to viscous damping or electrical resistance.
Synthesis: The three contributions are combined into a single energy balance equation. By dividing by the generalized flux $\dot{\phi}$ , the author derives a second-order linear differential equation.
Lagrangian Formulation: In the non-dissipative limit ( $R=0$ ), the author constructs a Lagrangian for the thermodynamic system, defining kinetic-like and potential-like terms, and establishing the corresponding canonical momentum and Hamiltonian.

Key Results
The derivation yields the following equation of motion for an open thermodynamic system:
$M \ddot{\phi} + R \dot{\phi} + K \phi = A(t)$
where:

$M$ is an effective thermodynamic inertial coefficient (derived from entropy flux dependence).
$R$ is an effective thermodynamic resistance (derived from Onsager coefficients for particle transfer).
$K$ is an effective thermodynamic restoring coefficient (derived from the pressure-volume response).
$A(t)$ is a generalized thermodynamic driving potential related to the rate of energy input ( $\dot{U}$ ).

The paper demonstrates that this equation is structurally identical to:

The mechanical equation: $m \ddot{x} + b \dot{x} + kx = F(t)$
The electrical equation: $L \ddot{q} + R_{el} \dot{q} + \frac{1}{C}q = V(t)$

Detailed dimensional analyses (provided in the appendices) confirm that the derived coefficients and variables possess consistent physical dimensions. Furthermore, the paper establishes a comprehensive correspondence table (Table I) mapping generalized coordinates, rates, coefficients, and driving terms across the three systems. The Lagrangian formulation (Table III) confirms that, in the non-dissipative limit, the thermodynamic system admits the same canonical structure as mechanical and electrical oscillators.

Significance and Claims
The paper claims that the oscillator-like structure of thermodynamic dynamics is not imposed by analogy with mechanical or electrical systems but emerges naturally from the first law of thermodynamics itself when subjected to dimensional reduction under near-equilibrium conditions.

Derivation vs. Analogy: The author emphasizes that the correspondence is a consequence of the derivation rather than an assumption. The inertial, dissipative, and restoring terms originate from distinct thermodynamic mechanisms (entropy deficit, irreversible particle transfer, and pressure response, respectively).
Universal Dynamical Structure: The work suggests that despite distinct physical origins, thermodynamic, mechanical, and electrical systems share a universal dynamical organization into inertial, dissipative, restoring, and driving contributions.
Scope and Limitations: The author modestly states that this result applies specifically to open systems within the near-equilibrium linear-response regime. The equation is not claimed to be a universal microscopic equation of motion for arbitrary systems far from equilibrium. Instead, it establishes that the first law contains sufficient structure to generate a mechanical-type dynamical representation once relevant near-equilibrium responses are identified.

The primary conceptual advance is the direct generation of a second-order equation of motion from the first law, providing a bridge between thermodynamic evolution and classical dynamical-systems theory without relying on pre-existing mechanical analogies.

Technical Summary: Equation of Motion for Open Thermodynamic Systems

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