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Dissipation
Dissipation
Dissipation
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In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, where the capacity of the final form to do thermodynamic work is less than that of the initial form. For example, transfer of energy as heat is dissipative because it is a transfer of energy other than by thermodynamic work or by transfer of matter, and spreads previously concentrated energy. Following the second law of thermodynamics, in conduction and radiation from one body to another, the entropy varies with temperature (reduces the capacity of the combination of the two bodies to do work), but never decreases in an isolated system.

In mechanical engineering, dissipation is the irreversible conversion of mechanical energy into thermal energy with an associated increase in entropy.[1]

Processes with defined local temperature produce entropy at a certain rate. The entropy production rate times local temperature gives the dissipated power. Important examples of irreversible processes are: heat flow through a thermal resistance, fluid flow through a flow resistance, diffusion (mixing), chemical reactions, and electric current flow through an electrical resistance (Joule heating).

Look up dissipation in Wiktionary, the free dictionary.

Definition

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Dissipative thermodynamic processes are essentially irreversible because they produce entropy. Planck regarded friction as the prime example of an irreversible thermodynamic process.[2] In a process in which the temperature is locally continuously defined, the local density of rate of entropy production times local temperature gives the local density of dissipated power.[definition needed]

A particular occurrence of a dissipative process cannot be described by a single individual Hamiltonian formalism. A dissipative process requires a collection of admissible individual Hamiltonian descriptions, exactly which one describes the actual particular occurrence of the process of interest being unknown. This includes friction and hammering, and all similar forces that result in decoherence of energy—that is, conversion of coherent or directed energy flow into an indirected or more isotropic distribution of energy.

Energy

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"The conversion of mechanical energy into heat is called energy dissipation." – François Roddier[3] The term is also applied to the loss of energy due to generation of unwanted heat in electric and electronic circuits.

Computational physics

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In computational physics, numerical dissipation (also known as "Numerical diffusion") refers to certain side-effects that may occur as a result of a numerical solution to a differential equation. When the pure advection equation, which is free of dissipation, is solved by a numerical approximation method, the energy of the initial wave may be reduced in a way analogous to a diffusional process. Such a method is said to contain 'dissipation'. In some cases, "artificial dissipation" is intentionally added to improve the numerical stability characteristics of the solution.[4]

Mathematics

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A formal, mathematical definition of dissipation, as commonly used in the mathematical study of measure-preserving dynamical systems, is given in the article wandering set.

Examples

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In hydraulic engineering

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Dissipation is the process of converting mechanical energy of downward-flowing water into thermal and acoustical energy. Various devices are designed in stream beds to reduce the kinetic energy of flowing waters to reduce their erosive potential on banks and river bottoms. Very often, these devices look like small waterfalls or cascades, where water flows vertically or over riprap to lose some of its kinetic energy.

Irreversible processes

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Important examples of irreversible processes are:

  1. Heat flow through a thermal resistance
  2. Fluid flow through a flow resistance
  3. Diffusion (mixing)
  4. Chemical reactions[5][6]
  5. Electrical current flow through an electrical resistance (Joule heating).

Waves or oscillations

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Waves or oscillations, lose energy over time, typically from friction or turbulence. In many cases, the "lost" energy raises the temperature of the system. For example, a wave that loses amplitude is said to dissipate. The precise nature of the effects depends on the nature of the wave: an atmospheric wave, for instance, may dissipate close to the surface due to friction with the land mass, and at higher levels due to radiative cooling.

History

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See also: Timeline of thermodynamics

The concept of dissipation was introduced in the field of thermodynamics by William Thomson (Lord Kelvin) in 1852.[7] Lord Kelvin deduced that a subset of the above-mentioned irreversible dissipative processes will occur unless a process is governed by a "perfect thermodynamic engine". The processes that Lord Kelvin identified were friction, diffusion, conduction of heat and the absorption of light.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Dissipation

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from Grokipedia
In physics, dissipation refers to the irreversible conversion of a system's usable , such as mechanical or , into that cannot be fully recovered for work, typically through processes like or viscous drag. This phenomenon is fundamental to the second law of , where it contributes to and the degradation of ordered into disordered . Dissipation occurs in diverse contexts, including mechanical systems where between surfaces dissipates kinetic as , and in where viscous forces convert into internal during flow. For instance, in a toy rolling down a track, is partially lost to -induced rather than fully converting to motion at the bottom. In electrical circuits, dissipation manifests as the transformation of into via resistance, quantified by the power dissipation formula P=I2RP = I^2 R, where II is current and RR is resistance. This process, analogous to mechanical friction, causes conductors to warm up and limits the of devices like resistors or wires. In , dissipation is tied to non-equilibrium processes, where the rate of loss to reflects the system's irreversibility and drives phenomena like . Examples include the heating of a gas during compression with frictional losses or the dissipation in ocean tides, estimated at 3.5 terawatts globally, primarily through viscous interactions in shallow seas. The study of dissipation is crucial for , such as minimizing losses in machines or predicting budgets in natural systems, and it underpins concepts like the fluctuation-dissipation theorem, which links random to systematic dissipation in equilibrium. Overall, dissipation highlights the inherent limitations of in practical applications, emphasizing the need for reversible processes to maximize utility.

Fundamentals

Physical Definition

In physics, dissipation refers to the irreversible transformation of structured, usable —such as mechanical or —into dispersed, less usable forms like , primarily through mechanisms including , electrical resistance, or . This process occurs when interactions within a convert ordered , associated with macroscopic motion, into random microscopic motion that cannot be fully recovered for work. A key physical mechanism of dissipation is evident in systems experiencing frictional forces, where a moving object gradually loses its as it slows down, with the lost energy appearing as due to collisions and vibrations at the molecular level between the object and its surroundings. For instance, a block sliding across a rough surface experiences drag from , which excites atomic vibrations in both the block and surface, heating them slightly while reducing the block's speed until it stops. An illustrative example is a simple released from an initial angle, where its initially alternates between at the bottom of the swing and at the extremes. However, air resistance acts as a dissipative proportional to the 's , converting portions of this into through viscous drag on the bob and string, causing the swing to decrease progressively until the rests motionless at the equilibrium position. This contrasts with reversible processes, which proceed without dissipation and can theoretically be undone without net loss or increase in disorder, such as an idealized in a with no oscillating perpetually. In real dissipative scenarios, however, the conversion always heightens disorder by spreading into unavailable thermal reservoirs.

Thermodynamic Context

In , dissipation serves as a fundamental manifestation of the second law, which asserts that all spontaneous processes are irreversible and result in a net increase in the of the . This irreversibility arises because dissipative mechanisms, such as or across gradients, transform mechanical or into in a disordered form, preventing the complete recovery of work. Consequently, the second law prohibits machines of the second kind, as any real process incurs dissipative losses that enforce the directionality of time through growth. Central to this context is the production of due to dissipation, which quantifies the degradation of available energy and the resultant loss of work potential. For an , the second is encapsulated by the inequality ΔS≥0\Delta S \geq 0, where ΔS\Delta S denotes the total entropy change and equality applies only to idealized reversible processes. To sketch the derivation, consider that entropy change for a reversible path is ΔS=∫dQrevT\Delta S = \int \frac{dQ_{\rm rev}}{T}, but dissipative irreversibilities introduce additional entropy generation; for instance, in QQ from a hot reservoir at ThT_h to a cold one at TcT_c, the universe's entropy change is ΔSuniv=−QTh+QTc>0\Delta S_{\rm univ} = -\frac{Q}{T_h} + \frac{Q}{T_c} > 0, with the positive term reflecting dissipation's contribution via non-quasistatic flow. This directly limits the of heat engines, where dissipative losses cap performance below the Carnot efficiency η=1−TcTh\eta = 1 - \frac{T_c}{T_h} by generating excess entropy that reduces net work output. Dissipation further embodies thermodynamic inefficiency by eroding —the maximum useful work extractable from a relative to its environment—in closed systems. Qualitatively, exergy loss occurs as dissipation increases internal disorder, converting high-quality into low-grade unavailable for work; this is formalized as the irreversibility I=T0ΔSgenI = T_0 \Delta S_{\rm gen}, where T0T_0 is the reference environmental temperature and ΔSgen\Delta S_{\rm gen} is the generated by dissipative processes. In essence, such losses diminish the system's capacity to perform work, underscoring dissipation's role in aligning all processes with law's mandate for universal ascent.

Mathematical Aspects

Dissipation in Dynamical Systems

In dynamical systems, dissipation is mathematically modeled through functions that capture energy loss mechanisms, particularly in Lagrangian formulations. The , introduced by Lord Rayleigh, provides a to represent velocity-dependent frictional forces. It is defined as Φ=12∑i∑jq˙iRijq˙j\Phi = \frac{1}{2} \sum_i \sum_j \dot{q}_i R_{ij} \dot{q}_j, where q˙i\dot{q}_i are the generalized velocities and RijR_{ij} is the symmetric positive semi-definite resistance (or damping) matrix encoding linear dissipative interactions between . This function enters the Euler-Lagrange equations via the generalized dissipative force Qjd=−∂Φ∂q˙jQ_j^d = -\frac{\partial \Phi}{\partial \dot{q}_j}, yielding the modified equations ddt(∂L∂q˙j)−∂L∂qj=Qjd+Qjext\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) - \frac{\partial L}{\partial q_j} = Q_j^d + Q_j^{ext}, where LL is the Lagrangian and QjextQ_j^{ext} are external non-conservative forces. Thus, the dissipation function systematically derives forces proportional to velocity, such as viscous drag, without explicit time-dependence in the Lagrangian. Dissipative dynamical systems are distinguished from conservative ones by their effect on phase space geometry, where volumes contract rather than remain preserved. In conservative systems, the flow is measure-preserving, satisfying ∇⋅f=0\nabla \cdot \mathbf{f} = 0 (Liouville's theorem), so infinitesimal phase space volumes evolve without change. In contrast, dissipative systems exhibit ∇⋅f<0\nabla \cdot \mathbf{f} < 0, leading to exponential contraction of phase space volumes at a rate given by the Lie derivative 1VdVdt=∇⋅f\frac{1}{V} \frac{dV}{dt} = \nabla \cdot \mathbf{f}, as seen in the damped pendulum where the contraction rate is −γ-\gamma for damping coefficient γ>0\gamma > 0. This contraction implies that trajectories converge toward lower-dimensional attractors, such as fixed points or limit cycles, while points outside these attractors belong to the wandering set, which is repelled and does not recur densely in any neighborhood. Lyapunov exponents quantify local expansion or contraction rates in , with negative values signaling dissipation. For a general x˙=f(x)\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}), the spectrum of Lyapunov exponents {λi}\{\lambda_i\} describes the average exponential rates of separation of infinitesimally close trajectories along directions of the ; the [sum ∑](/page/SumSum)λi=⟨∇⋅f⟩\sum](/page/Sum_Sum) \lambda_i = \langle \nabla \cdot \mathbf{f} \rangle the average phase space contraction rate, which is negative in dissipative systems. In one-dimensional systems, such as x˙=f(x)\dot{x} = f(x), the single Lyapunov exponent is λ=lim⁡t→∞1t∫0tf′(x(s)) ds\lambda = \lim_{t \to \infty} \frac{1}{t} \int_0^t f'(x(s)) \, ds, derived by linearizing the evolution of a perturbation δx˙=f′(x)δx\delta \dot{x} = f'(x) \delta x, yielding δx(t)≈δx(0)exp⁡(∫0tf′(x(s)) ds)\delta x(t) \approx \delta x(0) \exp\left( \int_0^t f'(x(s)) \, ds \right) and thus λ=lim⁡t→∞1tln⁡∣δx(t)/δx(0)∣\lambda = \lim_{t \to \infty} \frac{1}{t} \ln |\delta x(t)/\delta x(0)|. For a simple dissipative example like x˙=−γx\dot{x} = -\gamma x with γ>0\gamma > 0, f′(x)=−γf'(x) = -\gamma, so λ=−γ<0\lambda = -\gamma < 0, indicating uniform contraction toward the origin. A profound implication of dissipation in far-from-equilibrium dynamical systems is the emergence of dissipative structures, as conceptualized by . These are spatially or temporally organized states maintained by continuous energy and matter exchange with the environment, where dissipation drives self-organization rather than disorder. In systems driven far from equilibrium, instabilities amplify fluctuations, leading to bifurcations that form coherent patterns, such as Bénard convection cells, despite ongoing energy dissipation that would otherwise promote equilibrium uniformity. Prigogine's framework highlights how irreversibility in open systems enables complexity, with dissipation stabilizing these structures through feedback mechanisms like autocatalysis.

Numerical Dissipation in Computational Physics

In computational physics, numerical dissipation refers to the artificial damping of solution components introduced by discretization schemes when approximating partial differential equations (PDEs), particularly hyperbolic ones governing wave propagation and fluid flows. This dissipation arises inherently from methods like finite differences or finite volumes and can be intentionally added to enhance stability. For instance, in simulations of inviscid flows using the Euler equations, central difference schemes may produce non-physical oscillations near discontinuities such as shocks; to mitigate this, artificial dissipation is incorporated to suppress high-frequency modes while preserving overall accuracy. A common approach to introduce artificial dissipation is through upwind-biased schemes in computational fluid dynamics (CFD), which bias the stencil in the direction of wave propagation to stabilize solutions of hyperbolic PDEs. These schemes split the flux into antisymmetric (advective) and symmetric (dissipative) components, effectively adding a numerical viscosity that prevents oscillations without resolving physical viscous scales. For example, a third-order upwind-biased operator can be expressed as (δxu)j=16Δx(uj−2−6uj−1+3uj+2uj+1),(\delta_x u)_j = \frac{1}{6\Delta x} (u_{j-2} - 6u_{j-1} + 3u_j + 2u_{j+1}), where the symmetric part 112Δx(uj−2−4uj−1+6uj−4uj+1+uj+2)\frac{1}{12\Delta x} (u_{j-2} - 4u_{j-1} + 6u_j - 4u_{j+1} + u_{j+2}) contributes dissipation proportional to Δx3uxxxx/12\Delta x^3 u_{xxxx}/12. In Navier-Stokes simulations, artificial viscosity is often added explicitly as a term like ϵ∇2u\epsilon \nabla^2 \mathbf{u} (with ϵ\epsilon tuned to the grid scale) or in quadratic form q=ρcQΔx2(∂u/∂x)2q = \rho c_Q \Delta x^2 (\partial u / \partial x)^2 to smear shocks over a few cells, ensuring monotonicity and stability in under-resolved regions. This technique, originating from von Neumann and Richtmyer, mimics physical viscosity for shock capturing while avoiding excessive smoothing in smooth flow areas. Numerical errors in finite difference methods further manifest as phase errors and artificial damping, analyzed via dispersion relations that compare the numerical wavenumber-frequency relation to the exact PDE. For the one-dimensional advection equation ut+cux=0u_t + c u_x = 0, the exact dispersion relation is ω=ck\omega = c k (non-dispersive), but numerical schemes yield a complex ωn(k)\omega_n(k), where the imaginary part introduces damping (dissipation error from even-order truncation terms like ∂2u/∂x2\partial^2 u / \partial x^2) and the real part deviation causes phase errors (dispersion from odd-order terms like ∂3u/∂x3\partial^3 u / \partial x^3). In wave equations utt=c2uxxu_{tt} = c^2 u_{xx}, upwind schemes exhibit strong damping for short wavelengths (e.g., 2Δx\Delta x modes decay fully at CFL=0.5), while central schemes show dispersive trailing oscillations; stability requires ∣ωn∣≤∣ω∣|\omega_n| \leq |\omega| for all kk. To control these without excessive smearing, total variation diminishing (TVD) schemes, introduced by Harten, enforce TVD properties by limiting fluxes (e.g., using ϕ(r)\phi(r) limiters where 0≤ϕ≤20 \leq \phi \leq 2) to bound total variation while achieving second-order accuracy in smooth regions. In modern applications, managing numerical dissipation is crucial for accuracy in large-scale simulations. In climate modeling, large eddy simulations (LES) of boundary layers rely on monotone schemes that add dissipation, which combines with subgrid models; excessive dissipation (e.g., at low Smagorinsky constants Cs<0.3C_s < 0.3) leads to erroneous energy buildup or reduced fluxes, while optimal control ensures convergence independent of grid resolution. Similarly, in astrophysics, smoothed particle hydrodynamics (SPH) simulations of protoplanetary disks quantify dissipation via effective viscosity νeff\nu_{eff}, which depends on smoothing length hh and regularization cycles; values of νeff∼10−5\nu_{eff} \sim 10^{-5} (in code units) can damp waves if not tuned, but controlled dissipation improves shock resolution and multiphase flows without introducing unphysical diffusion. These contexts highlight numerical dissipation as both an artifact to minimize and a tool for robust, high-fidelity computations.

Engineering Applications

Hydraulic and Fluid Systems

In hydraulic and fluid systems, dissipation manifests primarily through frictional losses that convert mechanical energy into heat, impacting efficiency in pipelines, channels, and open flows. A key example is head loss in pipes, where viscous shear along the pipe walls dissipates kinetic energy. The Darcy-Weisbach equation quantifies this major head loss due to friction as hf=fLDv22gh_f = f \frac{L}{D} \frac{v^2}{2g}, where hfh_f is the head loss in meters, ff is the dimensionless friction factor, LL is the pipe length, DD is the diameter, vv is the average velocity, and gg is gravitational acceleration. The friction factor ff encapsulates the dissipative effects of wall roughness and flow regime, determined empirically via the Moody diagram or Colebrook-White equation for turbulent flows, and it scales the energy dissipation rate proportional to velocity squared, highlighting the nonlinear nature of frictional losses in engineering design. Viscosity plays a central role in dissipation within fluid dynamics, as captured by the Navier-Stokes equations, which include viscous stress terms that represent irreversible energy conversion to thermal form. The momentum equation for incompressible flow is ρ(∂u∂t+u⋅∇u)=−∇p+μ∇2u+ρg\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g}, where the term μ∇2u\mu \nabla^2 \mathbf{u} (with μ\mu as dynamic viscosity) accounts for diffusive momentum transfer leading to dissipation. This viscous dissipation is quantified by the dissipation function Φ=2μeijeij\Phi = 2\mu e_{ij} e_{ij}, where eije_{ij} is the strain rate tensor, representing the rate of mechanical energy converted to heat per unit volume. The scaling of dissipation varies with flow regime, governed by the Reynolds number Re=ρvDμRe = \frac{\rho v D}{\mu}: in laminar flows (Re<2300Re < 2300), dissipation is dominated by orderly viscous shearing with predictable parabolic velocity profiles, while in turbulent flows (Re>4000Re > 4000), enhanced mixing amplifies dissipation through chaotic eddies, often requiring empirical models like the friction factor in the Darcy-Weisbach equation. Engineers mitigate excessive dissipation in high-velocity flows using designed energy dissipators, such as stilling basins or baffle blocks in spillways, which intentionally induce to convert into and prevent downstream . Stilling basins create hydraulic jumps where supercritical flow abruptly transitions to subcritical, dissipating energy through intense and shear in the roller region, with design parameters like basin length scaled to the jump height based on . Baffles or chute blocks further enhance by obstructing flow, promoting energy loss via impact and formation, as standardized in U.S. Bureau of Reclamation guidelines for safe discharge at hydraulic structures. These structures ensure controlled dissipation, typically reducing from tens of meters per second to near-zero, thereby protecting while quantifying total loss through downstream depth measurements. In practical applications like systems, head loss directly translates to power dissipation, calculated as Ploss=ρgQhfP_{\text{loss}} = \rho g Q h_f, where ρ\rho is fluid density, QQ is , and hfh_f is the frictional head loss. For a typical with Q=10 m3/sQ = 10 \, \text{m}^3/\text{s}, hf=5 mh_f = 5 \, \text{m}, and (ρ=1000 kg/m3\rho = 1000 \, \text{kg/m}^3, g=9.81 m/s2g = 9.81 \, \text{m/s}^2), this yields Ploss≈0.5 MWP_{\text{loss}} \approx 0.5 \, \text{MW}, representing a significant fraction of gross potential power and underscoring the need for optimized pipe sizing to minimize viscous and turbulent losses.

Electrical and Electronic Systems

In electrical and electronic systems, dissipation primarily manifests as , where is converted into due to the resistance encountered by current flow. The power dissipated, denoted as PP, is given by P=I2[R](/page/R)P = I^2 [R](/page/R) or equivalently P=V2[R](/page/R)P = \frac{V^2}{[R](/page/R)}, where II is the current, VV is the voltage across the , and RR is the resistance. This phenomenon arises from the collisions of charge carriers with lattice ions and impurities, leading to irreversible generation that reduces system efficiency. In conductors like wires used in , resistive losses account for a significant portion of total dissipation, often comprising up to 5-10% of generated power in high-voltage lines. Semiconductors, such as in diodes and transistors, exhibit higher resistivity than metals, amplifying these losses, particularly under high current densities where local power densities can exceed 100 W/cm² in hotspots of high-performance integrated circuits. In integrated circuits (ICs), dissipation extends beyond steady-state conduction to include dynamic components like switching losses and leakage currents, necessitating advanced thermal management to prevent overheating and reliability degradation. Switching losses occur during transistor transitions between on and off states, where energy is dissipated as heat proportional to the switching frequency and voltage swing, often modeled as Psw=12CV2fP_{sw} = \frac{1}{2} C V^2 f, with CC as load capacitance and ff as frequency. Leakage currents, exacerbated by sub-10 nm scaling, contribute static power dissipation that increases exponentially with temperature, approximately doubling every 10°C and consuming 20-40% of total power in high-end processors as of 2020. A key figure of merit for assessing efficiency is the power efficiency η=PoutPout+Pdiss\eta = \frac{P_{out}}{P_{out} + P_{diss}}, where PoutP_{out} is the useful output power and PdissP_{diss} is the dissipated power; high-efficiency designs target η>90%\eta > 90\% to minimize thermal loads. Effective thermal management employs heat sinks, forced convection, or phase-change materials to maintain junction temperatures below 125°C, ensuring operational integrity in devices like CPUs and GPUs. Even in quantum-inspired electronic systems, dissipation is often analyzed through classical resistive models to approximate energy losses. In quantum dots, used in nanoscale transistors, charge transport involves tunneling with associated resistive dissipation, where effective resistance models predict heat generation similar to bulk semiconductors but scaled to picojoule levels per cycle. For superconductors, below the critical temperature, zero-resistance states eliminate ohmic losses, but any residual normal-state conduction introduces dissipation via parallel resistive channels, limiting applications in to cryogenic environments. To mitigate dissipative effects in high-power applications such as electric vehicles (EVs), engineers employ low-resistance materials like aluminum alloys or composites for interconnects, reducing I2RI^2 R losses by up to 30% compared to . Advanced cooling systems, including liquid immersion or thermoelectric coolers, further dissipate heat from power modules, enabling EV inverters to handle kilowatt-scale loads while keeping temperatures under 150°C and improving overall vehicle range by 5-10%.

Physical Phenomena

Irreversible Processes

Irreversible processes in provide fundamental examples of dissipation, where directed transfers or transformations degrade into disordered , rendering the changes non-reversible and increasing the total of the . These processes occur spontaneously in due to molecular-level interactions that favor equilibrium states, converting usable into without the possibility of complete recovery. Unlike reversible idealizations, real-world implementations involve inherent losses that exemplify the second law of , where is unavoidable. Heat conduction represents a classic dissipative , characterized by the spontaneous flow of from regions of higher to lower ones through a medium without bulk motion. This diffusion-driven transfer is quantified by Fourier's law, expressed as q=−k∇T\mathbf{q} = -k \nabla T, where q\mathbf{q} is the vector, kk is the material's conductivity, and ∇T\nabla T is the . The negative sign indicates flow opposite to the gradient, leading to equalization of and irreversible increase, as the process disperses organized thermal gradients into uniform without external work input. In natural settings, such as loss through , this dissipation prevents the reversal of temperature differences solely through conduction, highlighting the one-way nature of the spread. Mass diffusion similarly illustrates dissipation via irreversible mixing of substances, driven by concentration in a medium. Fick's first governs this process: J=−D∇c\mathbf{J} = -D \nabla c, where J\mathbf{J} is the diffusive , DD is the diffusion , and ∇c\nabla c is the concentration . The arises from the behavior of particles, akin to , where molecules move probabilistically, leading to net transport down the and eventual homogenization. This random dispersal converts stored in concentration differences into kinetic , producing as substances intermingle irreversibly, as seen in the spreading of solutes in aquifers. Non-equilibrium chemical reactions further demonstrate dissipation, particularly when barriers must be overcome, resulting in exothermic release that cannot be fully reversed. In , for instance, rapid oxidation reactions—such as the burning of hydrocarbons—transform into and products like and , with excess dissipating into the surroundings. These reactions occur far from equilibrium, where the forward pathway dominates due to high exothermicity, generating through the irreversible breakdown of ordered molecular structures into disordered gaseous states and . These processes occur dissipatively in , releasing that elevates local without recapturing the original chemical potentials. Viscous effects in fluid flows embody dissipation through internal friction among fluid particles, converting mechanical shear energy into heat in non-ideal, real-fluid dynamics. Unlike inviscid ideal flows, viscous dissipation arises from velocity gradients within the fluid, where adjacent layers slide against each other, generating frictional forces that degrade ordered motion. This process is prominent in laminar flows of viscous liquids like honey or in atmospheric boundary layers, where the energy loss manifests as a temperature rise and irreversible entropy production, preventing the fluid from returning to its initial kinetic state without external input. In oceanic currents, for example, tidal stirring induces such friction, dissipating tidal energy into widespread ocean warming.

Waves and Oscillations

In oscillatory systems, dissipation manifests as , which reduces the of oscillations over time by converting into or other forms. A fundamental example is the damped , modeled by the mx¨+bx˙+kx=0m \ddot{x} + b \dot{x} + k x = 0, where mm is , bb is the damping coefficient, kk is the spring constant, and dots denote time derivatives. The damping ratio ζ=b2mk\zeta = \frac{b}{2 \sqrt{m k}}
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