Revision #1185 → #1818 · back to history
modifiedEntropy (thermodynamic state variable)a17c30bdba29
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| note | Thermodynamic entropy as a physical state variable is not formalized in Mathlib; only information-theoretic and topological-dynamical entropies (e.g. `Real.negMulLog`, `coverEntropy`) exist. | Thermodynamic entropy as a physical state variable is not formalized in Mathlib; only information-theoretic and topological-dynamical entropies (e.g. `Real.negMulLog`) exist. |
modifiedEntropy change in a reversible processc6008ad44e60
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| anchors | [{"section":"Reversible process","snippet":"The entropy change [MATH] of a system can be well-defined as a small portion of heat"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} S={\\frac {\\delta Q_{\\mathsf {rev}}}{T}}}"}] | — |
modifiedCarnot cycle heat transfersed5ee3b093cb
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| anchors | [{"section":"Carnot cycle","snippet":"In a Carnot cycle, the heat [MATH] is transferred from a hot reservoir to a working gas at the constant temperature"},{"type":"math_alttext","value":"{\\displaystyle W={\\frac {T_{\\mathsf {H}}-T_{\\mathsf {C}}}{T_{\\mathsf {H}}}}\\cdot Q_{\\mathsf {H}}=\\left(1-{\\frac {T_{\\mathsf {C}}}{T_{\\mathsf {H}}}}\\right)Q_{\\mathsf {H}}}"}] | — |
modifiedCarnot's theorembe074e3020a8
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| anchors | [{"section":"Carnot cycle","snippet":"a heat engine with two thermal reservoirs can produce a work [MATH] if and only if there is a temperature difference between reservoirs"},{"type":"math_alttext","value":"{\\displaystyle W={\\frac {T_{\\mathsf {H}}-T_{\\mathsf {C}}}{T_{\\mathsf {H}}}}\\cdot Q_{\\mathsf {H}}=\\left(1-{\\frac {T_{\\mathsf {C}}}{T_{\\mathsf {H}}}}\\right)Q_{\\mathsf {H}}}"}] | — |
modifiedInternal energy as state function9494f1a46a21
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| anchors | [{"section":"Carnot cycle","snippet":"there exists a state function [MATH] with a change of [MATH] . It is called an internal energy"},{"type":"math_alttext","value":"{\\displaystyle W-Q_{\\Sigma }=W-\\left\\vert Q_{\\mathsf {H}}\\right\\vert +\\left\\vert Q_{\\mathsf {C}}\\right\\vert =W-Q_{\\mathsf {H}}-Q_{\\mathsf {C}}=0}"}] | — |
modifiedEntropy as state function (Clausius)7d478ecee107
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| anchors | [{"section":"Carnot cycle","snippet":"this equality implies existence of a state function [MATH] with a change of [MATH] and which is conserved over an entire cycle. Clausius called this state function entropy"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\left\\vert Q_{\\mathsf {H}}\\right\\vert }{T_{\\mathsf {H}}}}-{\\frac {\\left\\vert Q_{\\mathsf {C}}\\right\\vert }{T_{\\mathsf {C}}}}={\\frac {Q_{\\mathsf {H}}}{T_{\\mathsf {H}}}}+{\\frac {Q_{\\mathsf {C}}}{T_{\\mathsf {C}}}}=0}"}] | — |
modifiedTotal entropy change over Carnot cycle in reservoirs is zero6ca88aaebb55
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| anchors | [{"section":"Carnot cycle","snippet":"the total change of entropy in both thermal reservoirs over Carnot cycle is zero"},{"type":"math_alttext","value":"{\\displaystyle -{\\frac {Q_{\\mathsf {H}}}{T_{\\mathsf {H}}}}-{\\frac {Q_{\\mathsf {C}}}{T_{\\mathsf {C}}}}=\\Delta S_{\\mathsf {r,H}}+\\Delta S_{\\mathsf {r,C}}=0}"}] | — |
modifiedCarnot efficiency bound on less effective heat enginea004148073d6
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| anchors | [{"section":"Carnot cycle","snippet":"its work output is capped by Carnot efficiency"},{"type":"math_alttext","value":"{\\displaystyle W<\\left(1-{\\frac {T_{\\mathsf {C}}}{T_{\\mathsf {H}}}}\\right)Q_{\\mathsf {H}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {Q_{\\mathsf {H}}}{T_{\\mathsf {H}}}}+{\\frac {Q_{\\mathsf {C}}}{T_{\\mathsf {C}}}}<0}"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S_{\\mathsf {r,H}}+\\Delta S_{\\mathsf {r,C}}>0}"}] | — |
modifiedClausius equality48426d03adc5
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| anchors | [{"section":"Classical thermodynamics","snippet":"According to the Clausius equality , for a reversible cyclic thermodynamic process"},{"type":"math_alttext","value":"{\\displaystyle \\oint {\\frac {\\delta Q_{\\mathsf {rev}}}{T}}=0}"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} S={\\frac {\\delta Q_{\\mathsf {rev}}}{T}}}"}] | — |
modifiedEntropy via path-independent line integral57f1d5fbb577
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| anchors | [{"section":"Classical thermodynamics","snippet":"the line integral [MATH] is path-independent . Thus we can define a state function [MATH] , called entropy"},{"type":"math_alttext","value":"{\\displaystyle \\oint {\\frac {\\delta Q_{\\mathsf {rev}}}{T}}=0}"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} S={\\frac {\\delta Q_{\\mathsf {rev}}}{T}}}"}] | — |
modifiedBoltzmann/Gibbs entropy formula20941c8ee54b
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| anchors | [{"section":"Statistical mechanics","snippet":"entropy is a logarithmic measure for the system with a number of states, each with a probability"},{"type":"math_alttext","value":"{\\displaystyle S=-k_{\\mathsf {B}}\\sum _{i}{p_{i}\\ln {p_{i}}}}"}] | — |
modifiedContinuous-state entropy as expected log-probability0a63b63a6c1f
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| anchors | [{"section":"Statistical mechanics","snippet":"the summation is replaced by an integral over all possible states, or equivalently we can consider the expected value of the logarithm of the probability that a microstate is occupied"},{"type":"math_alttext","value":"{\\displaystyle S=-k_{\\mathsf {B}}\\left\\langle \\ln {p}\\right\\rangle }"},{"type":"math_alttext","value":"{\\displaystyle S=-k_{\\mathsf {B}}\\ \\mathrm {tr} {\\left({\\hat {\\rho }}\\times \\ln {\\hat {\\rho }}\\right)}}"}] | — |
modifiedVon Neumann entropy (density-matrix form)7d6973028429
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| anchors | [{"section":"Statistical mechanics","snippet":"where [MATH] is a density matrix , [MATH] is a trace operator and [MATH] is a matrix logarithm"},{"type":"math_alttext","value":"{\\displaystyle S=-k_{\\mathsf {B}}\\left\\langle \\ln {p}\\right\\rangle }"},{"type":"math_alttext","value":"{\\displaystyle S=-k_{\\mathsf {B}}\\ \\mathrm {tr} {\\left({\\hat {\\rho }}\\times \\ln {\\hat {\\rho }}\\right)}}"}] | — |
modifiedFundamental postulate of statistical mechanics63511fa4e9c0
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| anchors | [{"section":"Statistical mechanics","snippet":"among system microstates of the same energy (i.e., degenerate microstates ) each microstate is assumed to be populated with equal probability"},{"type":"math_alttext","value":"{\\displaystyle S=k_{\\mathsf {B}}\\ln {\\Omega }}"}] | — |
modifiedMicrocanonical ensembleac625b63d587
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| anchors | [{"section":"Statistical mechanics","snippet":"such a system is one with a fixed volume, number of molecules, and internal energy, called a microcanonical ensemble"},{"type":"math_alttext","value":"{\\displaystyle S=k_{\\mathsf {B}}\\ln {\\Omega }}"}] | — |
modifiedEquivalence of Gibbs and classical thermodynamic entropy7ba7dc0b34f7
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| anchors | [{"section":"Equivalence of definitions","snippet":"Proofs of equivalence between the entropy in statistical mechanics"},{"type":"math_alttext","value":"{\\displaystyle S=-k_{\\mathsf {B}}\\sum _{i}{p_{i}\\ln {p_{i}}}}"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} S={\\frac {\\delta Q_{\\mathsf {rev}}}{T}}}"}] | — |
modifiedFundamental thermodynamic relationad1c59f07eed
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| anchors | [{"section":"The fundamental thermodynamic relation","snippet":"This relation is known as the fundamental thermodynamic relation ."},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} U=T\\ \\mathrm {d} S-p\\ \\mathrm {d} V}"}] | — |
modifiedGibbs free energy change equation6c50793f4a41
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| anchors | [{"section":"Entropy in chemical thermodynamics","snippet":"this expression becomes the equation of Gibbs free energy change"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S_{\\mathsf {universe}}=\\Delta S_{\\mathsf {surroundings}}+\\Delta S_{\\mathsf {system}}}"},{"type":"math_alttext","value":"{\\displaystyle \\Delta G=\\Delta H-T\\ \\Delta S}"}] | — |
modifiedEntropy balance equation for open systems62ec377d283f
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| anchors | [{"section":"Entropy balance equation for open systems","snippet":"with respect to the rate of change with time [MATH] of the extensive quantity entropy [MATH] , the entropy balance equation is"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\mathrm {d} S}{\\mathrm {d} t}}=\\sum _{k=1}^{K}{{\\dot {M}}_{k}{\\hat {S}}_{k}+{\\frac {\\dot {Q}}{T}}+{\\dot {S}}_{\\mathsf {gen}}}}"}] | — |
modifiedEntropy generation equation (open-system second law)fcd97c1ff79f
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| anchors | [{"section":"Entropy balance equation for open systems","snippet":"the open system version of the second law is more appropriately described as the \"entropy generation equation\""},{"type":"math_alttext","value":"{\\displaystyle {\\dot {S}}_{\\mathsf {gen}}\\geq 0}"}] | — |
modifiedEntropy change for isothermal ideal gas expansione9e47b7da64d
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| anchors | [{"section":"Isothermal expansion or compression of an ideal gas","snippet":"For the expansion (or compression) of an ideal gas from an initial volume"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S=nR\\ln {\\frac {V}{V_{0}}}=-nR\\ln {\\frac {P}{P_{0}}}}"}] | — |
modifiedEntropy change at constant volume2b159b22dadd
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| anchors | [{"section":"Cooling and heating","snippet":"Similarly at constant volume, the entropy change is"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S=nC_{\\mathrm {V} }\\ln {\\frac {T}{T_{0}}}}"}] | — |
modifiedIdeal-gas total entropy change for varying T and V64f95f4d2a7f
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| anchors | [{"section":"Cooling and heating","snippet":"For an ideal gas, the total entropy change is"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S=nC_{\\mathrm {V} }\\ln {\\frac {T}{T_{0}}}+nR\\ln {\\frac {V}{V_{0}}}}"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S=nC_{\\mathrm {P} }\\ln {\\frac {T}{T_{0}}}-nR\\ln {\\frac {P}{P_{0}}}}"}] | — |
modifiedEntropy of fusionc2b89ac635e2
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| anchors | [{"section":"Phase transitions","snippet":"For fusion (i.e., melting ) of a solid to a liquid at the melting point [MATH] , the entropy of fusion is"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S_{\\mathsf {fus}}={\\frac {\\Delta H_{\\mathsf {fus}}}{T_{\\mathsf {m}}}}.}"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S_{\\mathsf {vap}}={\\frac {\\Delta H_{\\mathsf {vap}}}{T_{\\mathsf {b}}}}}"}] | — |
modifiedEntropy of vaporisation3dfc72f6e79a
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| anchors | [{"section":"Phase transitions","snippet":"for vaporisation of a liquid to a gas at the boiling point [MATH] , the entropy of vaporisation is"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S_{\\mathsf {fus}}={\\frac {\\Delta H_{\\mathsf {fus}}}{T_{\\mathsf {m}}}}.}"},{"type":"math_alttext","value":"{\\displaystyle \\Delta S_{\\mathsf {vap}}={\\frac {\\Delta H_{\\mathsf {vap}}}{T_{\\mathsf {b}}}}}"}] | — |
modifiedVon Neumann entropyed9d45ea646d
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| anchors | [{"section":"Entropy in quantum mechanics","snippet":"the concept of entropy was developed by John von Neumann and is generally referred to as"},{"type":"math_alttext","value":"{\\displaystyle S=-k_{\\mathsf {B}}\\ \\mathrm {tr} {\\left({\\hat {\\rho }}\\times \\ln {\\hat {\\rho }}\\right)}}"}] | — |
addedClassical limit of von Neumann entropyafe9a3370fa7
modifiedShannon entropyc14ddf5a908f
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| anchors | [{"section":"Information theory","snippet":"The definition of information entropy is expressed in terms of a discrete set of probabilities"},{"type":"math_alttext","value":"{\\displaystyle H(X)=-\\sum _{i=1}^{n}{p(x_{i})\\log {p(x_{i})}}}"}] | — |
modifiedBoltzmann entropy formula from Shannon1570b7026c5b
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| anchors | [{"section":"Information theory","snippet":"which is the Boltzmann entropy formula"},{"type":"math_alttext","value":"{\\displaystyle H=-\\sum _{i=1}^{W}{p_{i}\\ln {p_{i}}}=\\ln {W}}"},{"type":"math_alttext","value":"{\\displaystyle H=k\\ln {W}}"}] | — |
modifiedCalorimetric entropy measurement49e1536d8907
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| anchors | [{"section":"Measurement","snippet":"The measurement, known as entropymetry"},{"type":"math_alttext","value":"{\\displaystyle T:={\\left({\\frac {\\partial U}{\\partial S}}\\right)}_{V,N}\\ \\Rightarrow \\ \\cdots \\ \\Rightarrow \\ \\mathrm {d} S={\\frac {\\mathrm {d} Q}{T}}}"}] | — |