Legendre Transformation

6 Key excerpts on "Legendre Transformation"

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  An Introduction to Equilibrium Thermodynamics

  Pergamon Unified Engineering Series

  • Bernard Morrill, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Properly interpreted this means we have an equilibrium sys-tem with negligible kinetic and potential energies with respect to the center of mass of the system. Our thermodynamic model still allows for the change of composition within the system. It follows directly from Eq. (6.1.1) that by U=U(S,V,N i ); i= 1,2,... ,v (6.1.1) dU = dN t ; ΪΦ] (6.1.2) 190 Transformation of Thermodynamic Variables 191 It has been previously demonstrated that T(S,V,N t ) = (j[£) v ^ (6.1.3) -piS.VM-ffl)^ (6.1.4) and »«•wiwLj <61 ' 5) then dU = {T) v , N .dS -(p) SJIi dV+ Σ (H) s .v.s,dN t ; ι Φ j (6.1.6 ?= 1 Before we discuss the Legendre Transformation let us establish the entropie form of the fundamental equation: S = S ( l / , K , N , ) ; i = l , 2 , . . . , i ; (6.1.7) Then i*j (6.1.8) It can be shown that > -K ' ^ = ( i i , (β· 1 · 9 ) T< u > v >™ = {jp) 0JI{ t 6 -1 -1 0 ) and as a consequence

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  Combustion Thermodynamics and Dynamics

  • Joseph M. Powers(Author)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  In general, this is not true for arbitrary transformations. As discussed by Cantwell (2002), the Legendre Transformation for thermodynamic systems is a type of contact transformation discussed in the theory of Lie groups. EXAMPLE 3.5 Let y = y ( x 1 , x 2 , x 3 ) . With ψ i thus known to be ψ i = ∂y/∂x i , derive expressions for x 1 , ψ 2 , and ψ 3 in terms of ψ 1 , x 2 , and x 3 . Thus, replace x 1 as an independent variable by ψ 1 . Equivalently, replace ψ 1 as a dependent variable by x 1 . We have the associated differential form dy = ψ 1 dx 1 + ψ 2 dx 2 + ψ 3 dx 3 . (3.85) Because we know y ( x 1 , x 2 , x 3 ) , we know ψ i = ∂y/∂x i . Choose now a Legendre trans-formed variable τ 1 ≡ z ( ψ 1 , x 2 , x 3 ) : z = y − ψ 1 x 1 . (3.86) Then dz = ∂z ∂ψ 1 x 1 ,x 2 dψ 1 + ∂z ∂x 2 ψ 1 ,x 3 dx 2 + ∂z ∂x 3 ψ 1 ,x 2 dx 3 . (3.87) 4 Adrien-Marie Legendre (1752–1833), French mathematician. 5 Two differentiable functions f and g are said to be Legendre Transformations of each other if their first derivatives are inverse functions of each other: Df = ( Dg ) − 1 . With some effort, not shown here, one can prove that the Legendre Transformations of this section satisfy this general condition. 94 Mathematical Foundations of Thermodynamics Now, differentiating Eq. ( 3.86 ), one also gets dz = dy − ψ 1 dx 1 − x 1 dψ 1 . (3.88) Elimination of dy in Eq. ( 3.88 ) by using Eq. ( 3.85 ) gives dz = ψ 1 dx 1 + ψ 2 dx 2 + ψ 3 dx 3 dy − ψ 1 dx 1 − x 1 dψ 1 = − x 1 dψ 1 + ψ 2 dx 2 + ψ 3 dx 3 . (3.89) Thus, from Eq. ( 3.87 ), one gets x 1 = − ∂z ∂ψ 1 x 2 ,x 3 , ψ 2 = ∂z ∂x 2 ψ 1 ,x 3 , ψ 3 = ∂z ∂x 3 ψ 1 ,x 2 . (3.90) So, the original expression had three independent variables, x 1 , x 2 , x 3 , and three conjugate variables ψ 1 , ψ 2 , ψ 3 . Definition of the Legendre function 6 z with canonical variables ψ 1 , x 2 , and x 3 allowed determination of the remaining variables x 1 , ψ 2 , and ψ 3 in terms of the canonical variables.

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  Principles of Thermodynamics

  • Jean-Philippe Ansermet, Sylvain D. Brechet(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  76 Thermodynamic Potentials Equation (4.12) defines the Legendre transform of the state function F with respect to the state variable X i . To simplify the notation, we simply write, G = F − Y i X i (4.19) We obtain the state function H (Y 0 , Y 1 , . . . , Y n ) by performing successive Legendre trans- formations on the state function F (X 0 , X 1 , . . . , X n ), i.e. H = F − n  i=0 Y i X i (4.20) It is also possible to perform Legendre Transformations on a few variables only, which is often the case in thermodynamics. In the following section, we will obtain the thermodynamic potentials by performing several Legendre Transformations of the internal energy with respect to various extensive state variables. 4.3.1 Link between Thermodynamics and Mechanics In analytical mechanics, a Legendre Transformation defined with the opposite sign relates the Lagrangian L (r 1 , . . . , r n , v 1 , . . . , v n ) of a system containing n material points to the Hamiltonian H (r 1 , . . . , r n , P 1 , . . . , P n ) of this system [9], i.e. H = n  i=1 P i · v i − L (4.21) Since the Legendre transform of a state function is a state function, its opposite is also a state function and contains the same information. 4.4 Thermodynamic Potentials The extensive state function that plays a central role in thermodynamics is the internal energy U (S, V, {N A }). We call thermodynamic potential every state function obtained by a Legendre Transformation of U (S, V, {N A }). By extension, the internal energy itself can also be called a thermodynamic potential since it can be obtained by two successive Legendre Transformations starting from itself. The three other thermodynamic potentials that we will define in this section are the free energy, the enthalpy and the Gibbs free energy. 4.4.1 Free Energy The free energy F (T, V, {N A }), also called Helmholtz free energy, is defined as the Legendre transform (4.19) of the internal energy U (S, V, {N A }) with respect to the entropy S, i.e.

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  Molecular Engineering Thermodynamics

  • Juan J. de Pablo, Jay D. Schieber(Authors)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  3.1 Legendre transforms 55 Table 3.1 How to recover the fundamental energy relation U ( S , V ) from the generalized thermodynamic potentials F , G , and H as functions of their canonical independent variables. Given a fundamental relation of the quantity in the first column, use the equations in the second and third columns to eliminate the variables in the fourth column, leaving U = U ( S , V ). From Find canonical independent variables To find internal energy U ( S , V ) Eliminate H ( S , P ) V ( S , P ) = ( ∂ H /∂ P ) S , N U ( S , P ) = H ( S , P ) − PV ( S , P ) P F ( T , V ) S ( T , V ) = − ( ∂ F /∂ T ) V , N U ( T , V ) = F ( T , V ) + TS ( T , V ) T G ( T , P ) S ( T , P ) = − ( ∂ G /∂ T ) P , N U ( T , P ) = G ( T , P ) + TS ( T , P ) − PV ( T , P ) T , P V ( T , P ) = ( ∂ G /∂ P ) T , N From Eqns. ( 3.7 ) and ( 3.8 ) we can eliminate T to find U ( S ). Hence, knowing F ( T ) allows us to reconstruct the entire curve. Therefore, we define the Helmholtz potential F to be the Legendre transform of U ( S ) by Eqn. ( 3.6 ). Similarly, if we perform the Legendre transform on U ( V ), we obtain the enthalpy H as the Legendre transform; we can perform the Legendre transform simultaneously on both arguments of U ( S , V ) to obtain the Gibbs free energy G : H ( S , P , N 1 , . . . , N m ) : = U + PV , (3.9) F ( T , V , N 1 , . . . , N m ) : = U − TS , (3.10) G ( T , P , N 1 , . . . , N m ) : = U − TS + PV . (3.11) Note that the sign is different for the transformation from V to P , because the slope of U ( V ) is − P . For each of the three transforms, we can reconstruct the original fundamental energy relation by Eqns. ( 3.7 ) and ( 3.8 ). The results of the inversion are summarized in Table 3.1 . For a given general potential, only one set of independent variables gives complete thermodynamic information. We call the members of this set the canonical independent variables for that potential.

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  Methods of Mathematical Physics, Volume 2

  Partial Differential Equations

  • Richard Courant, David Hilbert(Authors)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  A similar important application of the Legendre Transformation occurs in fluid dynamics': Steady flow of a two-dimensional com- pressible fluid is described by two velocity components u, v as func- tions of the rectangular coordinates x, y. Suppose the sound speed c is a given function of u2 + z?. The motion is governed by the first order system of equations u, - V , = 0, 2 ( C 2 - U )U, - UZ1(U, + V,) + (C2 - V 2 ) V , = 0. Accordingly, there exists a velocity potential 4(x, ZJ) such that and 11 = +I, v = 4 u (c2 - 4f)+zz - 2 4 Z # J U 4 , , + (CZ - 4h#Ju,, = 0. A crucial step in dealing with this nonlinear differentmid equation of second order is the Legendre transformat ion 9 + = u2 + 7 y , 42 = 71, = V , 9, = 2, R = y, See also Chapt,er V, and R . Courant, and K . 0. Friedrichw, 11 1, 1111 247-249. CAUCHY-KOWALEWSKY EXISTEIGCE PROOF 39 It, yields for +(u, 21) : L liiwar tliffcrciitial equation of sccoiid order (C2 - 1i2)@?,?, + 21/1@u, + (c‘ - V’2)@‘tr,r = 0 which is iiseful for solving inaiiy flov problcnis.’ 57. l h e Exislerace ’f’lteorenc of Cuuchy and hbzuuleztdiy We conclude this chapter with the discussion of a fundanwiital theorem w-hich assures the existence of solutions of partial differential equations and at the same time clarifies the manlier in which arbitrary functions enter into the “general” solution. The theorem is due to Cauchy, who initiated the modern theory of partial differential equations. Sophie Kowalewski in her Thesis, inspired by Weierstrass, has carried out the proof in a rather general manner.2 The theorem refers to the “initial value problem” as or we often shall call it “Cauchy’s problem”.

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  Advanced Thermodynamics Engineering

  • Kalyan Annamalai, Ishwar K. Puri, Milind A. Jog(Authors)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  285 Chapter 5 Postulatory (Gibbsian) Thermodynamics Objectives ◾ Postulatory (mathematical) approach to thermodynamics ◾ Introduce Legendre transforms and relate to thermodynamic variables ◾ Thermodynamic postulates (or statements of Laws of Thermodynamics in mathematical forms) 5.1 Introduction In the previous chapters we discussed the thermodynamics laws by employing a classical approach. In this chapter we will discuss the subject using a set of postulates or rules, fundamental state equations, and other mathematical tools such as Legendre transforms. We will first establish the classical rationale behind such an approach and relate it to some postulates (without invoking any laws). Thereafter, we will introduce the Legendre transform, which is possible to transfer an equation from one coordinate system to others (e.g., from an entropy-based coordinate to a temperature-based coordinate). Next, we will relate the energy to work. We will discuss the pos-tulates in a mathematical context, present the entropy and energy fundamental equations, and describe intensive and extensive properties by using the properties of homogeneous functions. Finally, we will derive the Gibbs–Duhem relation using fundamental and Euler equations. 5.2 Classical Rationale for Postulatory Approach We have seen that a stable equilibrium state (SES) is achieved when the entropy reaches a maximum value for fixed values of U, V, and m (or for fixed number of moles N 1 , N 2 , … ). The 286 ◾ Advanced Thermodynamics Engineering, Second Edition internal energy of an open system that exchanges species with its surroundings or a reacting sys-tem is represented by the relation U = U(S, V, N 1 , N 2 , N 3 , …), (5.1) which is also known as the energy fundamental equation. The internal energy is a single valued function of S, V, N 1 , N 2 , N 3 , … , since there is a single SES for a specified set of conditions.

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