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Entropy production

Entropy production is the irreversible generation of entropy within a thermodynamic system arising from processes such as friction, viscous dissipation, heat conduction across finite temperature differences, and chemical reactions, serving as a quantitative measure of the departure from reversibility as mandated by the second law of thermodynamics.^[1] This production is inherently non-negative, equaling zero only in idealized reversible processes and positive otherwise, reflecting the unavoidable dissipation of useful work into heat and the consequent increase in the total entropy of the system and its surroundings.^[1] In mathematical terms, the local entropy production rate, often denoted as \sigma, is defined as the time derivative of the internal entropy per unit volume due to irreversible mechanisms, \sigma = \frac{d_i s}{dt} \geq 0, where s is the entropy density (internal entropy per unit volume).^[1] In the framework of non-equilibrium thermodynamics, entropy production plays a central role in analyzing open systems far from equilibrium, where continuous exchanges of energy and matter with the environment sustain steady states characterized by positive but minimized entropy generation.^[2] Developed extensively by Ilya Prigogine, this field employs the bilinear form of entropy production as \sigma = \sum J_k X_k, summing over thermodynamic fluxes J_k (e.g., heat flux) conjugate to thermodynamic forces X_k (e.g., temperature gradients), which ensures \sigma > 0 and underpins principles like minimum entropy production for linear regimes near equilibrium.^[2] Beyond classical applications in engineering—such as assessing efficiency losses in heat engines or fluid flows—entropy production extends to complex phenomena, including self-organization in dissipative structures like Bénard cells or biological metabolism, where local entropy increase paradoxically enables global order in far-from-equilibrium conditions.^[2]^[3]

Introduction and Fundamentals

Definition and basic principles

Entropy production, often denoted as σ, represents the rate at which entropy is generated within a thermodynamic system due to irreversible processes. In an isolated system, where no matter or energy is exchanged with the surroundings, the entropy production is defined as the time derivative of the total entropy, σ = dS/dt ≥ 0, ensuring that the entropy can only increase or remain constant over time.^[4] This quantity captures the fundamental irreversibility inherent in natural thermodynamic processes, distinguishing them from idealized reversible ones. Reversible processes, which occur infinitely slowly and maintain the system in equilibrium at every stage, exhibit zero entropy production (σ = 0), as there is no net generation of entropy from internal mechanisms. In contrast, irreversible processes, such as those involving friction, heat conduction across finite temperature differences, or unrestricted expansion, result in positive entropy production (σ > 0), leading to an overall increase in the system's entropy.^[5] This distinction underscores the directional nature of thermodynamic evolution toward equilibrium states. The total change in entropy for any thermodynamic system can be expressed as the sum of exchanged and internally produced contributions:

dS = dS_e + dS_i,

where dS_e accounts for the entropy transferred to or from the surroundings (typically \delta Q / T for heat exchange at temperature T), and dS_i is the internal entropy production, which satisfies dS_i \geq 0 with equality only for reversible processes.^[5] In rate form, the production term relates to σ, emphasizing that dS_i = \sigma \, dt. Entropy production serves as a quantitative measure of energy dissipation and the associated loss of useful work potential, often termed "lost work," where the dissipated energy is proportional to T \sigma.^[4] This concept aligns with the second law of thermodynamics by quantifying the inevitable degradation of energy quality in real-world processes.

Relation to thermodynamic laws

Entropy production arises from the integration of the first law of thermodynamics, which expresses energy conservation as dU = \delta Q + \delta W, with the second law, which states that the entropy of an isolated system cannot decrease, dS \geq 0.^[6] In this framework, the second law is expressed through the Clausius inequality for a cyclic process: \oint \frac{\delta Q}{T} \leq 0, where equality holds for reversible processes and the inequality reflects irreversibility due to internal dissipation.^[6] Combining these laws for a closed system undergoing infinitesimal changes yields the entropy balance equation dS = \frac{\delta Q}{T} + d_i S, where d_i S \geq 0 is the irreversible entropy change, or production, arising from dissipative processes within the system.^[6] To derive the explicit form of entropy production, consider the first law alongside the definition of entropy for reversible processes, T dS = \delta Q_\text{rev}. For irreversible cases, the actual heat transfer \delta Q differs from \delta Q_\text{rev}, leading to T dS = \delta Q + T \sigma \, dt, where \sigma \geq 0 is the local entropy production rate per unit volume, and \delta Q includes contributions from heat and work.^[6] Rearranging with the first law and assuming only pressure-volume work \delta W = -P dV, the combined expression becomes

T dS = dU + P dV + T \sigma \, dt,

with the term T \sigma \, dt quantifying the irreversible dissipation that ensures the second law's entropy increase.^[6] This form highlights entropy production as the bridge between conserved energy (first law) and directional entropy growth (second law), applicable under the local thermodynamic equilibrium assumption where properties like temperature are defined locally despite gradients.^[6] The Clausius inequality generalizes to non-equilibrium conditions by decomposing the total entropy change into exchanged and produced parts: \Delta S \geq \int \frac{\delta Q}{T}, with equality for reversible paths and the difference \int \frac{\delta Q}{T} - \Delta S \leq 0 representing integrated entropy production over the process.^[6] In non-equilibrium thermodynamics, this extends to open systems with flows, where the production rate \sigma is expressed as \sigma = \sum_k J_k X_k \geq 0, with J_k as thermodynamic fluxes (e.g., heat or matter flow) and X_k as conjugate forces (e.g., temperature or chemical potential gradients), ensuring the second law holds locally.^[6] This formalism was foundationalized by Onsager, who in 1931 derived reciprocal relations for linear non-equilibrium processes near equilibrium, linking fluxes and forces via J_i = \sum_j L_{ij} X_j with symmetric coefficients L_{ij} = L_{ji}, derived from microscopic reversibility and applied to entropy production minimization in stationary states.^[7] These relations establish entropy production as a bilinear form that governs coupled transport phenomena, such as thermoelectric effects, while upholding the second law through positive definiteness of the L_{ij} matrix.^[7]

Historical Development

Early concepts

The foundational ideas of entropy production emerged in the early 19th century through efforts to understand the efficiency limits of heat engines and the inherent waste in thermal processes. Sadi Carnot, in his 1824 publication Reflections on the Motive Power of Fire, analyzed ideal reversible engines operating between heat reservoirs, emphasizing that maximum work extraction requires infinitesimal temperature differences to avoid losses. He explicitly recognized irreversibilities in real-world operations, such as friction in mechanical parts, which dissipates motive power into heat, and uneven heat conduction across finite temperature gradients, which similarly degrades available energy without producing useful work. Carnot's insight—that these dissipative effects represent a fundamental barrier to perfect efficiency—laid the groundwork for quantifying energy unavailability, though he framed it within caloric theory rather than modern energy conservation.^[8] Rudolf Clausius advanced these concepts in the 1850s by integrating the conservation of energy with Carnot's principles, establishing entropy as a key measure of process inefficiency. In his 1850 memoir On the Moving Force of Heat, Clausius introduced the idea of a state function that tracks the transformation of heat into work, showing that for reversible cycles, the integral of heat transferred divided by temperature equals zero, while irreversible processes yield a positive value, indicating entropy increase. By 1865, in The Mechanical Theory of Heat, he formalized entropy (denoted S) as this state function, whose change in any process satisfies an inequality: equality holds for reversible paths, but real cycles produce excess entropy due to dissipative mechanisms like friction and spontaneous heat flow. This inequality encapsulated the second law's directive that entropy production signifies the degradation of usable energy in isolated systems.^[8] Lord Kelvin (William Thomson) complemented Clausius's work in the mid-19th century by focusing on the practical implications of dissipation for energy availability. In his 1851 paper On the Dynamical Theory of Heat, Kelvin articulated the second law as the impossibility of converting heat entirely into work without compensatory effects elsewhere, introducing the notion of "unavailable energy"—that portion of thermal energy rendered useless for work due to irreversible dissipation. He illustrated this through examples like heat conduction in solids, where temperature equalization dissipates potential without external compensation, and friction in machinery, which converts mechanical energy into dispersed heat. Kelvin's 1848 proposal of an absolute temperature scale, grounded in Carnot efficiency, further highlighted how dissipation accumulates over time, leading to a universal tendency toward energy degradation in finite systems.^[8] By the late 19th century, these classical insights shifted thermodynamic inquiry from idealized equilibrium cycles toward the realities of ongoing irreversible processes, prompting early explorations beyond static state functions. Pioneers like James Clerk Maxwell and Ludwig Boltzmann began incorporating kinetic theory to model dissipative phenomena such as viscosity and diffusion, revealing that entropy production arises continuously in non-isolated systems interacting with surroundings. This transition underscored the limitations of equilibrium-focused thermodynamics, setting the stage for analyzing dynamic, far-from-equilibrium behaviors without yet formalizing a comprehensive non-equilibrium framework.^[8]

Key contributions in non-equilibrium thermodynamics

In the early 20th century, Lars Onsager made foundational contributions to non-equilibrium thermodynamics by deriving reciprocal relations that connect phenomenological coefficients in the linear regime to entropy production. In his 1931 papers, Onsager demonstrated that for systems near equilibrium, the fluxes of irreversible processes, such as heat conduction and diffusion, are linearly related to thermodynamic forces, with the matrix of coefficients being symmetric, ensuring the entropy production remains positive definite.^[9] These relations, grounded in the principle of microscopic reversibility, provided a rigorous framework for quantifying entropy generation in open systems, influencing subsequent developments in transport theory. Building on Onsager's work, Ilya Prigogine advanced the understanding of entropy production in the 1940s and 1950s through his formulation of local entropy production and the principle of minimum entropy production for steady states near equilibrium. In his 1947 study, Prigogine showed that the rate of entropy production in a system can be expressed as a sum of local contributions from irreversible processes, such as heat and matter flows, allowing for a bilinear form that highlights the dissipative nature of non-equilibrium states.^[2] By the 1950s, he established that in linear irreversible thermodynamics, steady-state solutions minimize the total entropy production under fixed boundary conditions, a variational principle that stabilizes near-equilibrium dynamics while underscoring the second law's role in open systems.^[10] This approach shifted focus from global to local descriptions, enabling analysis of dissipative structures in chemical reactions and fluid dynamics. The 1960s and 1970s saw the emergence of extended irreversible thermodynamics (EIT), which addressed limitations of classical theory by incorporating fast-relaxing variables, such as viscous fluxes and heat fluxes, directly into the entropy function to model systems farther from equilibrium. Pioneered by Ingo Müller in his 1967 entropy balance formulation, EIT extends the Gibbs relation to include non-equilibrium contributions, yielding hyperbolic transport equations with finite propagation speeds, unlike the parabolic ones in standard thermodynamics. Subsequent developments by Müller and collaborators in the 1970s, including collaborations with Tommaso Ruggeri, applied EIT to relativistic fluids and polymers, demonstrating how entropy production governs relaxation times and non-Fourier heat conduction, thus bridging microscopic relaxation to macroscopic irreversibility. This framework proved essential for describing rapid transients in materials science and plasma physics. Post-2000 research has integrated entropy production concepts with complexity theory, particularly in far-from-equilibrium systems, revealing how dissipation drives self-organization and emergent behaviors. The maximum entropy production principle (MEPP), formalized by Roderick Dewar in 2003 and reviewed by Martyushev and Seleznev in 2006, posits that in steady states, systems maximize entropy production subject to constraints, contrasting Prigogine's minimum principle and explaining complexity in biological and geophysical flows through variational optimization. More recently, Jeremy England's 2013 work on dissipation-driven adaptation links high entropy production rates to the selection of self-replicating structures in non-equilibrium environments, providing a thermodynamic basis for evolutionary complexity in chemical and biological systems. These integrations highlight entropy production as a selector for complex, adaptive dynamics in open systems, with applications from atmospheric circulation to molecular evolution.

Irreversible Processes and Examples

Common examples of irreversibility

One prominent example of irreversibility arises in mechanical systems where friction converts ordered mechanical energy into disordered thermal energy, thereby producing entropy. In such processes, the dissipative work done by friction generates heat that increases the entropy of the system and its surroundings, in accordance with the second law of thermodynamics. This entropy production is inherent to the viscous dissipation within the material, where molecular interactions resist relative motion and lead to a net increase in thermal disorder.^[11]^[12] Uncontrolled expansion of gases provides another classic illustration of entropy production, as seen in free expansion where a gas suddenly expands into a vacuum without performing work or exchanging heat. In this adiabatic process, the internal energy remains constant for an ideal gas, but the volume increase leads to a positive entropy change due to the greater number of accessible microstates, reflecting the irreversible mixing of the gas with the empty space. The Joule-Thomson effect, involving throttling through a porous plug, similarly demonstrates irreversibility for real gases, where pressure drop at constant enthalpy results in entropy generation from intermolecular forces and non-uniform flow.^[13]^[14]^[15] In chemical reactions proceeding away from equilibrium, entropy production occurs through the chemical affinity, which drives the reaction toward completion by favoring the direction that minimizes free energy. The affinity, defined as the negative sum of chemical potential differences times stoichiometric coefficients, quantifies the thermodynamic force, and its coupling with reaction rate yields a positive entropy production rate, ensuring the process's spontaneity. This dissipation manifests as heat release or other irreversible transformations, distinguishing non-equilibrium reactions from reversible ones at equilibrium.^[16]^[17] Diffusion across concentration gradients exemplifies irreversibility when solute particles spread spontaneously from high to low concentration regions without external work, leading to entropy production via the homogenization of the system. This process, governed by Fick's laws, involves random molecular motions that increase the positional disorder, with the entropy generation arising from the flux-affinity product in non-equilibrium thermodynamics. Unlike controlled diffusion in membranes that might extract work, uncontrolled cases purely dissipate the gradient's potential energy as thermal entropy.^[18]

Entropy production in heat transfer and fluid flow

In heat transfer processes, entropy production arises primarily from thermal gradients driving irreversible heat conduction, as described by Fourier's law. The local heat flux \mathbf{q} is given by \mathbf{q} = -\kappa \nabla T, where \kappa is the thermal conductivity and \nabla T is the temperature gradient. The corresponding volumetric entropy production rate \sigma due to this conduction is \sigma = \frac{\kappa}{T^2} (\nabla T)^2, where T is the absolute temperature.^[19] This expression, derived from the bilinear form of fluxes and forces in linear irreversible thermodynamics, quantifies the irreversibility as heat flows from higher to lower temperatures, generating entropy at a rate proportional to the square of the gradient.^[20] In fluid flows, entropy production is also significant due to viscous dissipation, which converts mechanical energy into heat through internal friction. For Newtonian fluids governed by the Navier-Stokes equations, the viscous stress tensor \boldsymbol{\tau} relates to the rate-of-strain tensor, and the dissipation function \Phi captures the work done against viscous forces. The entropy production rate from viscosity is \sigma_v = \frac{\eta}{T} \Phi, where \eta is the dynamic viscosity and \Phi = 2 \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \frac{\partial u_i}{\partial x_j} / 2 (summation implied), representing the squared velocity gradients.^[19] This term highlights how shear and extensional flows, such as in pipe or channel flows, inevitably produce entropy, limiting the efficiency of fluid transport systems.^[20] When heat and mass transfer occur simultaneously in non-isothermal flows, entropy production becomes more complex, involving coupled fluxes. In such systems, the total entropy production includes contributions from thermal conduction, viscous dissipation, and diffusive mass transfer, often expressed as \sigma = \frac{\kappa}{T^2} (\nabla T)^2 + \frac{\eta}{T} \Phi - \sum_k \frac{J_k \cdot \nabla \mu_k}{T}, where J_k and \mu_k are the mass flux and chemical potential of species k.^[20] These couplings, analyzed through Onsager reciprocal relations, show that temperature variations can enhance or suppress mass diffusion, leading to higher overall irreversibility in processes like drying or combustion. In engineering applications, such as boundary layers over surfaces, entropy production integrates both conduction and viscous effects, providing insights into aerodynamic drag and heat transfer efficiency. For instance, in a laminar boundary layer, the entropy generation peaks near the wall due to high velocity and temperature gradients, with total production scaling as \int \sigma \, dV \propto Re^{-1/2}, where Re is the Reynolds number, emphasizing the trade-off between friction and heat losses. This analysis aids in optimizing designs like turbine blades or heat exchangers by minimizing localized dissipation.

Thermodynamic Devices and Efficiency

Heat engines

In the Carnot cycle, which serves as the theoretical benchmark for heat engines operating between a hot reservoir at temperature T_h and a cold reservoir at T_c, all processes are reversible, resulting in zero net entropy production (\sigma = 0). This idealization assumes infinitesimal temperature differences during heat transfer, absence of friction, and quasi-static operations, allowing the engine to achieve the maximum possible efficiency \eta = 1 - T_c / T_h without generating entropy.^[21] Real heat engines, however, inevitably produce entropy (\sigma > 0) due to practical irreversibilities such as mechanical friction in moving parts and finite-rate heat transfer across temperature gradients. Friction dissipates mechanical energy into heat, increasing the entropy of the system, while finite heat transfer rates—necessary for finite-time operation—create temperature drops between reservoirs and the working fluid, leading to non-quasistatic processes that further elevate entropy production. These factors reduce the engine's efficiency below the Carnot limit, with entropy generation directly quantifying the thermodynamic losses.^[22]^[23] For specific cycles like the Otto and Diesel engines used in internal combustion, entropy production arises from both heat transfer and internal processes. The total entropy production is given by

\sigma_\text{total} = \int \frac{\delta Q_h}{T_h} - \int \frac{\delta Q_c}{T_c},

where the integrals represent contributions during heat addition (\delta Q_h at boundary temperature T_h) and rejection (\delta Q_c at T_c), accounting for irreversibilities in heat transfer and internal processes through the temperatures and heats. In the Otto cycle, constant-volume heat addition amplifies entropy production due to rapid temperature rises, while the Diesel cycle's constant-pressure combustion introduces additional losses from incomplete mixing and heat losses. These contributions limit efficiencies to around 30-40% in practice, far below Carnot values.^[23] Endoreversible engine models address finite-time constraints by assuming reversible internal cycles but irreversible heat exchanges with reservoirs, attributing entropy production solely to conductive heat transfer across finite temperature differences. Optimizing for maximum power yield the Curzon-Ahlborn efficiency \eta_\text{CA} = 1 - \sqrt{T_c / T_h}, which incorporates the entropy generated during non-equilibrium heat flows and provides a closer match to observed engine performances than the reversible Carnot efficiency. This approach highlights how minimizing entropy production in finite-time operations balances power output against thermodynamic ideals.^[24] The broader consequence of entropy production in heat engines is the reduction in extractable work, formalized as the lost work W_\text{lost} = T_0 \sigma, where T_0 is the ambient temperature serving as the reference environment. This term represents the exergy destruction due to irreversibilities, directly linking entropy generation to the unavailable energy that could otherwise contribute to useful output, and underscores the second law's constraint on engine performance.^[25]

Refrigerators and heat pumps

Refrigerators operate by extracting heat from a cold reservoir at temperature T_c and rejecting it to a hot reservoir at T_h > T_c, requiring work input, while heat pumps reverse this process to deliver heat to the hot reservoir. The ideal reversible Carnot refrigerator achieves the maximum coefficient of performance (COP), defined as the ratio of heat extracted from the cold reservoir to the work input, given by \mathrm{COP} = \frac{T_c}{T_h - T_c}, with zero entropy production due to the absence of irreversibilities.^[26] In practice, irreversibilities such as finite-rate heat transfer and internal friction generate entropy \sigma > 0, reducing the COP below the Carnot limit; for an endo-irreversible model, the COP becomes \mathrm{COP} = \frac{\Delta S_C T_c}{T_h - T_c + \Delta S_C (T_h / T_c) + \Delta S_I T_h}, where \Delta S_C is the entropy change at the cold side and \Delta S_I is the internal entropy production.^[26] The vapor-compression cycle, widely used in practical refrigerators, introduces significant entropy production primarily through the throttling process in the expansion valve, which is an isenthalpic expansion causing a temperature drop but generating entropy due to the lack of work recovery, and through heat transfer irreversibilities in the evaporator and condenser arising from finite temperature differences (often termed heat leaks in design contexts). Entropy production in this cycle is calculated from exergy destruction contributions in each stage, such as compressor inefficiency (I_{\mathrm{comp}}), condenser (I_{\mathrm{cond}} = C_p (T_3 - T_2) - T_0 C_p \ln(T_3 / T_2)), throttling (I_{\mathrm{exp}} = C_p (T_4 - T_3) - T_0 C_p \ln(T_4 / T_3)), and evaporator (I_{\mathrm{evp}} = C_p (T_1 - T_4) - T_0 C_p \ln(T_1 / T_4)), with T_0 as the reference temperature and C_p the specific heat; total entropy generation is \sigma = I_t / T_0, where I_t is the total exergy destruction.^[27] These irreversibilities can reduce the COP by up to 20-30% compared to ideal values, depending on refrigerant and operating conditions, such as R134a systems showing \sigma \approx 533 \, \mathrm{J/kg \cdot K} at minimum.^[27] Heat pumps function as reversed Carnot engines, transferring heat from a cold external source to a warm interior space, with the ideal heating COP \mathrm{COP}_h = \frac{T_h}{T_h - T_c}, but real entropy production from similar sources lowers this metric. Seasonal performance factors, such as the heating seasonal performance factor (HSPF) or seasonal coefficient of performance (SCOP), integrate entropy production effects over varying ambient conditions and load profiles, yielding average efficiencies for adsorption systems where \sigma minimization reduces losses by balancing pressure drops and heat exchanger areas.^[28] In adsorption heat pumps, for instance, normalized entropy production correlates inversely with SCOP.^[28] The principle of minimum entropy production guides optimal design of refrigerators and heat pumps by allocating heat conductances and minimizing \sigma across components, often yielding the highest COP for given constraints; for example, in endo-irreversible refrigerators, optimal allocation satisfies G_h / G_c = \sqrt{(T_h - T_c) / T_c}, where G are conductances, differing from power maximization criteria and ensuring \sigma does not exceed levels that degrade performance beyond 10-15%.^[26] This approach, rooted in finite-time thermodynamics, has been applied to compression-resorption heat pumps to equalize local \sigma in absorbers through staged processes.^[29]

Power dissipation in systems

In systems where electrical or mechanical power is dissipated, entropy production arises from irreversible processes that convert ordered energy into thermal disorder. Power dissipation, such as through resistive heating or frictional losses, generates heat that increases the entropy of the system and its surroundings, quantifying the degradation of available work potential. This phenomenon is central to understanding inefficiencies in devices like electronic circuits and mechanical actuators, where the rate of entropy production \sigma is directly tied to the dissipated power divided by the local temperature T. A primary example is Joule heating in electrical resistors, where current flow through a resistance R produces heat at a rate I^2 R, leading to entropy production \sigma = \frac{I^2 R}{T}. This expression derives from the second law applied to ohmic losses, where the electrical work is irreversibly converted to thermal energy, increasing entropy without reversible recovery. In circuit analysis, the total entropy production across multiple components sums these contributions, often manifesting as the dominant source of irreversibility in low-voltage electronics. The associated exergy loss, which measures the destroyed useful work, equals T \sigma, linking dissipation directly to thermodynamic inefficiency via the Gouy-Stodola theorem. Mechanical power dissipation occurs in systems with friction, such as sliding contacts or bearings, where the frictional force f opposing motion at velocity v dissipates power f \cdot v, yielding entropy production \sigma = \frac{f \cdot v}{T}. This formulation captures the conversion of mechanical work into heat through viscous or dry friction, a process inherently irreversible due to non-equilibrium contact dynamics. In engineering contexts, such losses are prevalent in rotating machinery, where they compound with electrical dissipation to limit overall performance. Applications of these principles extend to electronics and electric motors, where combined Joule heating in windings and frictional losses in rotors contribute significantly to entropy production, often exceeding 20-30% of input power as thermal waste in typical designs. In thermoelectric devices, entropy production due to coupled electrical and thermal currents further complicates efficiency, as Peltier and Seebeck effects generate additional irreversible heat fluxes that elevate \sigma beyond simple resistive terms. Minimizing these dissipations through low-friction coatings has become a focus for enhancing device longevity and energy utilization.

Mathematical Expressions and Formulations

General expressions for entropy production

In non-equilibrium thermodynamics, the local rate of entropy production \sigma(\mathbf{x}, t) in a continuous system describes the irreversible generation of entropy due to processes such as heat conduction and diffusion. This quantity is fundamentally expressed as a bilinear form involving thermodynamic fluxes and their conjugate forces:

\sigma(\mathbf{x}, t) = \sum_k J_k(\mathbf{x}, t) \, X_k(\mathbf{x}, t),

where the J_k represent the fluxes (e.g., heat flux \mathbf{J}_q or diffusive mass flux \mathbf{J}_i) and the X_k are the associated thermodynamic forces (e.g., temperature gradient \nabla(1/T) or affinity gradients related to chemical potential differences). This formulation arises from the balance equation for entropy in open systems, ensuring a positive definite structure that aligns with the second law. The bilinear form guarantees that \sigma(\mathbf{x}, t) \geq 0 at every point \mathbf{x} and time t, reflecting the local manifestation of the second law of thermodynamics: entropy can only increase or remain constant locally due to irreversible processes, with equality holding only in equilibrium states where all fluxes and forces vanish. This pointwise non-negativity is a cornerstone of the theory, distinguishing it from equilibrium thermodynamics where entropy changes are solely due to reversible exchanges.^[30] In the linear regime near equilibrium, where forces are small, the fluxes are linearly related to the forces via phenomenological coefficients:

J_i = \sum_j L_{ij} X_j,

with the Onsager reciprocal relations imposing symmetry L_{ij} = L_{ji}. These relations, derived from microscopic reversibility, ensure the bilinear form yields a positive semi-definite entropy production, \sigma = \sum_{i,j} L_{ij} X_i X_j \geq 0. The symmetry has profound implications for cross-effects in coupled transport phenomena. For a finite system, the total entropy production rate \Pi(t) is obtained by integrating the local production over the volume V:

\Pi(t) = \int_V \sigma(\mathbf{x}, t) \, dV \geq 0.

This integral quantifies the overall irreversibility within the system, with the total entropy balance including both production and exchange terms. In steady states, \Pi remains constant and positive, highlighting the persistence of dissipation.^[31]

Specific cases: heat flow and mass diffusion

In linear non-equilibrium thermodynamics, the entropy production due to heat flow arises from the coupling between the heat flux and the temperature gradient, as part of the general bilinear form of fluxes and thermodynamic forces. For uncoupled heat conduction, the local entropy production rate \sigma is expressed as

\sigma = \mathbf{J}_q \cdot \nabla \left( \frac{1}{T} \right),

where \mathbf{J}_q is the heat flux vector and T is the absolute temperature.^[20] This form ensures \sigma \geq 0, reflecting the second law, since \mathbf{J}_q flows from higher to lower temperature while \nabla (1/T) points in the same direction. According to Fourier's law, \mathbf{J}_q = -\kappa \nabla T, with \kappa > 0 the thermal conductivity coefficient. Substituting yields

\sigma = \frac{\kappa (\nabla T)^2}{T^2},

a quadratic expression that quantifies dissipation, scaling with the square of the temperature gradient and inversely with T^2. This derivation holds under near-equilibrium conditions, where gradients are small compared to microscopic scales.^[20] For mass diffusion, entropy production stems from concentration gradients driving particle fluxes in multi-component systems. The contribution to \sigma is

\sigma = -\sum_i \mathbf{J}_i \cdot \nabla \left( \frac{\mu_i}{T} \right),

where \mathbf{J}_i is the diffusion flux of species i and \mu_i its chemical potential.^[20] In isothermal conditions (\nabla T = 0), this simplifies to \sigma = -\sum_i (\mathbf{J}_i / T) \cdot \nabla \mu_i, with positivity ensured because \mathbf{J}_i opposes \nabla \mu_i. Fick's first law relates the flux to concentration c_i via \mathbf{J}_i = -D_i \nabla c_i, where D_i > 0 is the diffusion coefficient. For ideal dilute solutions, \nabla \mu_i = (RT / c_i) \nabla c_i, leading to

\sigma = RT \sum_i \frac{D_i (\nabla c_i)^2}{c_i T},

or \sigma = R \sum_i D_i (\nabla c_i)^2 / c_i, which highlights how steeper concentration gradients or lower concentrations amplify irreversibility. This form applies to binary or multi-component diffusion without external forces.^[20] Coupled transport phenomena extend these expressions when heat and mass flows interact with other processes, such as charge transport. In thermoelectric effects, temperature gradients induce electric fields (Seebeck effect), and electric currents carry heat (Peltier effect), captured by cross-coefficients in the linear phenomenological equations \mathbf{J}_q = L_{qq} \nabla (1/T) + L_{qe} \mathbf{E}/T and \mathbf{J}_e = L_{eq} \nabla (1/T) + L_{ee} \mathbf{E}/T, where \mathbf{E} = -\nabla \phi is the electric field and L_{ij} are Onsager coefficients with L_{qe} = L_{eq} by reciprocity. The total entropy production becomes \sigma = \mathbf{J}_q \cdot \nabla (1/T) + \mathbf{J}_e \cdot (-\nabla (\phi / T)), incorporating both direct and coupled dissipations; for example, the Peltier coefficient \Pi = L_{qe} / L_{ee} relates heat transported per unit charge. Electrokinetic effects, like those in charged membranes, analogously couple ion diffusion to pressure or electric gradients, with similar bilinear forms involving \nabla ( \tilde{\mu}_i / T ), where \tilde{\mu}_i includes electrochemical potential. These couplings enable devices like thermocouples, where entropy production balances efficiency gains from cross-effects.^[20] For large gradients where linear approximations break down—such as in rapid transients or nanoscale systems—non-linear extensions modify the entropy production framework. Extended irreversible thermodynamics treats fluxes like \mathbf{J}_q as independent state variables, incorporating relaxation times \tau and yielding a generalized entropy s = s_{le} - (\tau \mathbf{J}_q^2)/(2 \lambda T^2), where s_{le} is the local-equilibrium entropy and \lambda = \kappa T^2 relates to conductivity. The production rate then includes non-linear terms, such as \sigma = \mathbf{J}_q \cdot \nabla (1/T) - (\tau / T) \mathbf{J}_q \cdot (d\mathbf{J}_q / dt) + \mathbf{J}_q^2 / (\lambda T), accounting for flux relaxation and hyperbolic heat conduction equations like \tau \partial_t \mathbf{J}_q + \mathbf{J}_q = -\kappa \nabla T. This avoids paradoxes like infinite propagation speeds in Fourier's law and applies to non-Fourier behaviors in polymers or biological tissues, with entropy production remaining non-negative but dependent on flux magnitudes.

Specific cases: mixing and expansion processes

In the irreversible mixing of ideal gases at constant temperature and pressure, the process occurs spontaneously when two or more gases are allowed to intermingle without external work or heat exchange beyond equilibration, leading to an increase in the system's entropy. This entropy production arises from the loss of distinguishability between the separate components and the expansion into a larger volume, reflecting the second law of thermodynamics. For an ideal mixture, the total entropy change, which equals the entropy production σ since the system is isolated, is given by

\Delta S_\text{mix} = -n R \sum_i x_i \ln x_i,

where n is the total number of moles, R is the gas constant, x_i = n_i / n is the mole fraction of component i, and the sum is over all components.^[32] This formula quantifies the configurational disorder introduced by mixing, with the negative sign indicating the logarithmic form's origin in probabilistic considerations.^[33] The Gibbs paradox emerges when applying this formula to mixing identical gases, where the predicted entropy increase persists even as the gases become indistinguishable, contradicting the extensivity of entropy. The resolution in classical statistical mechanics lies in treating particles of the same species as indistinguishable from the outset, introducing the 1/N! factor in the partition function, which eliminates the spurious entropy change for identical gases while preserving it for distinct ones.^[34] This adjustment ensures thermodynamic consistency without altering the mixing entropy for truly different ideal gases.^[32] Free expansion, or Joule expansion, involves an ideal gas expanding into a vacuum within an insulated container, where no work is done and no heat is exchanged, yet the process is irreversible due to the absence of equilibrium during expansion. The entropy production is the system's entropy change, calculated via a reversible isothermal path between initial and final states, yielding

\sigma = n R \ln \left( \frac{V_f}{V_i} \right),

with V_f and V_i the final and initial volumes, respectively; temperature remains constant for an ideal gas.^[35] This positive value demonstrates dissipation through volume increase without compensating mechanisms.^[35] In irreversible adiabatic expansion with external work, such as a gas pushing a piston against constant external pressure in an insulated setup, the process generates entropy through non-quasistatic pressure differences. The first law gives ΔU = -P_ext (V_f - V_i), leading to a temperature drop, and the entropy production is

\sigma = n C_v \ln \left( \frac{T_f}{T_i} \right) + n R \ln \left( \frac{V_f}{V_i} \right),

where C_v is the molar heat capacity at constant volume, and T_f is determined from the energy balance; unlike reversible adiabatic expansion, σ > 0 due to irreversibility.^[35] This highlights how finite pressure gradients produce entropy beyond mere state changes.

Microscopic and Stochastic Perspectives

Microscopic interpretation

In statistical mechanics, the macroscopic concept of entropy production finds its microscopic foundation in the irreversible dynamics of molecular systems, where irreversibility arises from the vast number of microscopic configurations consistent with macroscopic observables. At this level, entropy production quantifies the tendency toward equilibrium through processes like collisions or transitions that reduce the distinguishability of microstates, as formalized in the kinetic theory of gases and beyond.^[36] A cornerstone of this interpretation is Boltzmann's H-theorem, which demonstrates the monotonic increase of entropy in dilute gases described by the Boltzmann equation. The H-function, defined as H = \int f(\mathbf{v}) \ln f(\mathbf{v}) \, d\mathbf{v}, where f(\mathbf{v}) is the velocity distribution function, evolves according to \frac{dH}{dt} = -\sigma, with \sigma \geq 0 representing the entropy production rate due to binary collisions. This relation links entropy production directly to the contraction of accessible phase space under the molecular chaos assumption, ensuring that deviations from equilibrium diminish over time. The fluctuation-dissipation theorem further connects microscopic entropy production to thermal fluctuations in near-equilibrium systems. In linear response theory, the theorem relates the dissipative response of a system to external perturbations—manifesting as entropy production—to the equilibrium correlations of fluctuations, such that the transport coefficients governing dissipation are proportional to the integral of time correlation functions of the fluctuating variables. This establishes that irreversible entropy generation is intrinsically tied to the reversible fluctuations inherent in microscopic dynamics. For systems modeled by master equations governing Markovian transitions between discrete states, the microscopic entropy production rate adopts a specific bilinear form. It is given by \sigma = \sum_{i < j} (p_i w_{ij} - p_j w_{ji}) \ln \frac{p_i w_{ij}}{p_j w_{ji}}, where p_i is the probability of state i, and w_{ij} is the transition rate from i to j. This expression captures the irreversibility arising from asymmetric transition rates, quantifying how broken detailed balance at the molecular level drives net entropy increase.^[36] Coarse-graining provides another key perspective, bridging microscopic details to macroscopic observations by averaging over fine-grained variables, which introduces an additional information-theoretic component to entropy production. In this framework, the total entropy production decomposes into a physical part from underlying dynamics and an information entropy term reflecting the loss of detail upon projection onto coarser descriptions; the latter ensures overall non-negativity while allowing apparent violations at reduced resolutions. This aligns with the general macroscopic expressions for entropy production but highlights how microscopic irreversibility persists through informational erasure.

Entropy production in stochastic thermodynamics

In stochastic thermodynamics, entropy production is analyzed for small-scale systems exhibiting fluctuations, such as colloidal particles or biomolecules, governed by equations like the overdamped Langevin equation or the corresponding Fokker-Planck equation. This framework extends classical thermodynamics to individual stochastic trajectories, allowing the definition of thermodynamic quantities like work, heat, and entropy production on a trajectory-by-trajectory basis. The total entropy production along a trajectory, denoted as \Delta s_{\text{tot}}, quantifies the irreversibility of the process and is always non-negative on average, reflecting the second law for nonequilibrium systems in contact with thermal baths.^[37] The total entropy production decomposes into two parts: \Delta s_{\text{tot}} = \Delta s_{\text{med}} + \Delta s_{\text{sys}}, where \Delta s_{\text{med}} is the entropy change in the surrounding medium and \Delta s_{\text{sys}} is the change in the system's nonequilibrium entropy. The medium entropy production \Delta s_{\text{med}} arises from heat exchange with the bath and, for an isothermal process at temperature T, is given by \Delta s_{\text{med}} = Q / T, with Q the heat absorbed by the system along the trajectory; its sign is opposite for the bath. For a single particle in an overdamped regime driven by a conservative force F(x) = -\partial_x U(x), this is expressed as \Delta s_{\text{med}} = \frac{1}{T} \int_0^\tau F(x_t) \circ d x_t, where the Stratonovich product accounts for the stochastic integral along the trajectory. The system entropy change \Delta s_{\text{sys}} captures the evolution of the probability distribution and includes boundary terms \ln [p(x(0),0)/p(x(\tau),\tau)] plus an integral term involving the logarithmic derivative of the probability density along the path, ensuring \Delta s_{\text{tot}} satisfies detailed fluctuation relations even under time-dependent driving.^[37] A cornerstone result is the integral fluctuation theorem for the total entropy production over a finite time \tau: \langle e^{-\Delta s_{\text{tot}}} \rangle = 1, where the average is over an ensemble of trajectories starting from arbitrary initial distributions. This theorem implies that while \Delta s_{\text{tot}} can be negative for individual trajectories (allowing apparent violations of the second law), such events are exponentially suppressed, with the average \langle \Delta s_{\text{tot}} \rangle \geq 0. Derived for both continuous Langevin dynamics and discrete master equations, it generalizes earlier equalities like Jarzynski's work relation and holds for finite-time processes, providing a rigorous link between microscopic fluctuations and macroscopic irreversibility. In the long-time limit, it connects to the microscopic H-theorem for the decay of the probability distribution toward steady states.^[37] Applications of these concepts have been pivotal in understanding molecular machines and biological systems since the 2010s, where fluctuations dominate. For instance, in ATP-powered motors like kinesin, stochastic thermodynamics quantifies the minimal entropy production required for directed motion against loads, revealing efficiencies up to 60% but with trade-offs dictated by fluctuation theorems. In biological processes such as ion pumps or DNA replication, the framework assesses the thermodynamic cost of precision, showing that error correction in protein synthesis incurs excess dissipation bounded by information-theoretic limits derived from entropy production rates. These insights, validated experimentally via optical tweezers and single-molecule tracking, have illuminated how living systems operate near fundamental thermodynamic bounds while harnessing noise for function.

Advanced Topics and Applications

Equivalence to other thermodynamic formulations

Entropy production serves as a fundamental measure of irreversibility in thermodynamic processes, equivalent to the destruction of exergy or the loss of available work. In exergy analysis, the rate of exergy destruction is directly proportional to the entropy production rate, given by \dot{\Psi}_{destroyed} = T_0 \sigma, where T_0 is the ambient temperature and \sigma is the entropy production rate; this quantifies the degradation of energy's ability to perform useful work due to irreversibilities such as friction or heat transfer across finite temperature differences.^[38] The Gouy-Stodola theorem formalizes this equivalence, stating that the total lost work W_{lost} over a process equals the ambient temperature times the total entropy generated: W_{lost} = T_0 \Delta S_{gen}, emphasizing that entropy production represents the portion of energy rendered unavailable for work.^[39] In infinitesimal terms, the lost work increment is \delta W_{lost} = T_0 \sigma \, dt, linking local irreversibilities to the immediate dissipation of work potential.^[40] This connection extends to finite-time thermodynamics, where processes occur over constrained durations, and optimization balances power output against entropy production to enhance efficiency. In such frameworks, the ecological optimization criterion maximizes the function \dot{W} - T_0 \sigma, where \dot{W} is the power; this approach yields operating conditions with reduced entropy production compared to maximum power alone, achieving a trade-off that minimizes environmental impact while sustaining viable performance in heat engines or refrigerators.^[41] For instance, applying this to an irreversible Carnot engine demonstrates that ecological optimization lowers the entropy production rate by balancing thermal conductances and time allocations, outperforming endoreversible models in practical sustainability metrics.^[42] Modern extensions bridge entropy production to information theory, particularly through the Landauer principle, which equates the thermodynamic cost of computation to irreversible entropy generation. The principle asserts that erasing one bit of information in a system at temperature T produces at least k_B \ln 2 of entropy in the environment, corresponding to a minimum dissipated energy of k_B T \ln 2, where k_B is Boltzmann's constant; this establishes entropy production as the physical limit for logically irreversible operations like data erasure in computational devices.^[43] This thermodynamic-information equivalence underscores how entropy production governs the energy overhead in information processing, influencing designs in nanoscale electronics and quantum computing where minimizing dissipation aligns with reducing computational entropy generation.^[44]

Inequalities, stability, and homogeneous systems

The second law of thermodynamics, when extended to nonequilibrium processes, manifests as the inequality for the local entropy production rate, \sigma \geq 0, where equality holds only at thermodynamic equilibrium when all fluxes vanish. This inequality ensures that irreversible processes generate entropy, preventing perpetual motion and dictating the direction of spontaneous evolution in isolated systems. In open systems maintained away from equilibrium, \sigma > 0 quantifies the dissipation inherent to steady-state maintenance.^[2]^[45] Prigogine's minimum entropy production theorem applies to steady states in systems sufficiently close to equilibrium, stating that under fixed boundary conditions, the entropy production \sigma reaches a minimum value, with the first variation \delta \sigma = 0 for admissible perturbations consistent with the constraints. This variational principle, derived within linear irreversible thermodynamics where phenomenological coefficients are constant, characterizes stable steady states by minimizing dissipation relative to time-dependent trajectories under the same conditions. The theorem holds for systems where the linear response regime applies, such as those with small deviations from equilibrium, and it provides a criterion for selecting the steady state among possible configurations.^[2]^[20] These inequalities underpin stability criteria in nonequilibrium thermodynamics, particularly through the Glansdorff-Prigogine evolution theorem, which posits that the excess entropy production \delta_2 \sigma (second-order variation) is non-positive near a steady state, ensuring the system's return to stability after small perturbations. In this framework, positive \sigma during evolution indicates an approach toward the steady state, as the system minimizes dissipation, acting akin to a Lyapunov function that decreases over time. This stability is robust in linear regimes but can break down far from equilibrium, leading to bifurcations; however, within the theorem's scope, it guarantees asymptotic stability for constrained systems.^[2] In homogeneous systems, lacking spatial gradients, entropy production arises solely from internal processes like chemical reactions, where \sigma > 0 drives the system toward equilibrium without external fluxes. For instance, in a uniform reacting mixture, the positive entropy production from reaction affinities ensures monotonic approach to chemical equilibrium, serving as a stability indicator as concentrations evolve to minimize \sigma. This analysis highlights how, even in spatially uniform setups, the second law enforces directional stability through dissipative mechanisms.^[2]^[45]

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