Yasar demirel, in nonequilibrium thermodynamics third edition, 2014. However, while the fourier transform consists of an integration with a kernel, the legendre transform uses maximization as the transformation procedure. The legendre transform is commonly used in upper division and graduate physics courses, especially in classical mechanics,1 statistical mechanics, and thermodynamics. Legendre transformation an overview sciencedirect topics. Legendre transformation explained with animation youtube. This is a general feature of legendre transformations.
Addison february 25, 2003 introduction in this lecture we introduce other thermodynamic potentials and maxwell relations. In thermodynamics, the internal energy u can be legendre transformed into various thermodynamic potentials, with associated conjugate pairs of. The legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Legendre transforms mark alford, 20190215 1 introduction to legendre transforms if you know basic thermodynamics or classical mechanics, then you are already familiar with the legendre transformation, perhaps without realizing it. Legendre transform is shown on 2 in fact, the legendre transform of f looks as shown on graph 2. The legendre transform is a useful device for producing new state functions. When a legendre transform of a thermodynamic potential is defined, a partial derivative of that thermodynamic potential is introduced as a new variable. V an p eter and antali m at e february, 20 contents 1 introduction 2 1. Pdf legendre transforms in nonequilibrium thermodynamics.
Taking the differentials of the seven thermodynamic potentials defined in eqs. Use of legendre transforms in chemical thermodynamics. Pdf use of legendre transforms in chemical thermodynamics. Alberty and others published use of legendre transforms in chemical thermodynamics find, read and cite all the research you need on researchgate. The transform is well behaved only if fx is a convex function. Legendre transformation in more than one dimension for a differentiable realvalued function on an open subset u of r n the legendre conjugate of the pair u, f is defined to be the pair v, g, where v is the image of u under the gradient mapping d f, and g is the function on v. These relationships are often highlighted by the mathematical technique, legendre transformations 1,2. A legendre transform is a change in natural variables that is accomplished by defining a new thermodynamic potential by subtracting from the internal energy or.
The main application of and the historical root of the notion of legendre transform in differential geometry is in classical physics and its formalization by symplectic geometry. A legendre transform is a procedure for expressing the information content of some function by. Jan 10, 20 trying an example from a textbook but i dont understand legendre transform at all. In thermodynamics, it is usual to employ a slight variant, di ering by a sign from the usual legendre transform. The legendre transform is a mathematical tool with many applications in classical mechanics 1, 2, statistical mechanics, thermodynamics 3, 4, and computer vision problems 5. Legendre transform in twodimensional thermodynamics haynes miller fall, 2008 the legendre transform is a useful device for producing new state functions from old ones. In thermodynamics, the internal energy u can be legendre transformed into various. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by fourier transforms. Pdv, there are two natural variables and there are 2 2. The legendre transform is a transformation from a convex differentiable function fx to a function that depends on the family of tangents s. Legendre transformations many important thermodynamic equations are closely related. In thermodynamics, we demonstrate how the lt arising naturally from statistical mechanics and show how use of dimensionless thermodynamic potentials lead to more natural and symmetric relations.
Examples of the legendre transform relationship in physics. You may consider the legendre transformation to be a mere redefinition of variables but thats why its so important in practice. The legendrefenchel transform generalizes the legendre transform for the nondifferentiable or nonconvex functions, by finding a onetoone correspondence between the independent variable x and the derivative of fx. For the involution transform commonly used in classical mechanics and thermodynamics, see legendre transformation. Legendre transforms in nonequilibrium thermodynamics. Legendre transform an overview sciencedirect topics. Legendre transforms introduce other set of natural variables. Martinberz, in advances in imaging and electron physics, 1999. The legendre transformation connects two ways of specifying the same physics, via functions of two related. The entire legendretransform structure of thermodynamics can be obtained using fisher information in place of boltzmanns entropy. Pdf thermodynamics, statistical mechanics and entropy. Some of them play a supplemental role, while the others, such as the bessel and legendre functions, are of primary importance.
Symmetric representation of the legendre transform this symmetric geometrical construction allows us to display a number of useful and elegant relations that shed light on the workings of the legendre transform. Fishers information i allows then for such a construction. Legendre transformation and thermodynamics physics stack. This is illustrated by the elimination of ph in the calculation of standard gibbs. The generalization of the legendre transformation to affine spaces and nonconvex functions is known as the convex conjugate also called the legendrefenchel transformation, which can be used to construct a functions convex hull.
Topics include ensembles, fermi, bose and boltzmann statistics, density matrices, mean. The legendre transformation is a useful mathematical tool that is used in thermodynamics, classical mechanics and quantum field theory. The energy and entropy representations we have noted that both su,v,nand us,v,ncontain complete thermodynamic information. With reference to thermodynamics, callen 3 discusses application of legendre transformations. Thermodynamic potentials and maxwells relations stephen r. Manogue, david roundy, and elizabeth gire 1department of physics, oregon state university, corvallis, or 97331 one commonly and persistently di cult topic in thermodynamics is the relationship between.
Jun 06, 2008 the legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. The understanding of the difference between the two transforms is important be9 cause the general transform which arises in statistical mechanics is the legendre fenchel 10. Since the legendre transformation is selfinverse, it is interesting to study what will happen if it is applied to the hamiltonian. Pdf making sense of the legendre transform joe redish. The use of legendre transforms is not restricted to thermodynamics. How, therefore, might one transform the internal energy function, u, to provide an alternative thermodynamic function by replacing one, or more, of its extensive. Use of legendre transforms in chemical thermodynamics 51 biochemical applications may involve the coupling of reactions and mechanical work or the coupling of reactions with the transport of ions between phases at different electric potentials. This article is about an integral transform using legendre polynomials. Kennerly, a graphical derivation of the legendre transform, pdf online at. A legendre transform has made a major contribution in mechanics. This abstract legendre structure constitutes an essential ingredient that allows one to build up a statistical mechanics. The legendre transform is an encoding of the convex hull of a functions epigraph in terms of its supporting hyperplanes.
In the 2nd lecture, we will discuss the mathematics of thermodynamics, i. The legendre fenchel transform generalizes the legendre transform for the nondifferentiable or nonconvex functions, by finding a onetoone correspondence between the independent variable x and the derivative of fx. It is commonly used in classical mechanics to derive the hamiltonian formalism out of the lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of. The precise way in which the legendre fenchel transform generalizes the 7 legendre transform is carefully explained and illustrated with many examples and pic8 tures. Thermodynamics, statistical mechanics and entropy article pdf available in entropy 1911. A legendre transform converts from a function of one set of variables to another function of a conjugate set of variables. Dec 14, 2018 a look at how to get enthalpy from our fundamental combined first and second law.
A legendre transformation is hence a variable transformation. To change to another set of natural variables, a new function is defined by subtracting off a particular conjugate pair. You have to remember that the statements that helmholtz free energy is minimized at equilbrium and gibbs free energy is minimized at equilbrium are referring to two different notions of minimizing. Legendre transform of function 1 shown on graph 1 endpoints corresponding to the lines in fx. Me346a introduction to statistical mechanics wei cai. Besides information geometry, legendre transformation plays a fundamental role in formulating problems of thermodynamics, and in hamiltonlagrange. William young 18631942 was an english mathematician who made signi. Maybe the most famous application is that in classical mechanics, quantum mechanics and quantum field theory the hamiltonian and the lagrangian are connected by a legendre transformation. In classical mechanics, the hamiltonian function h h is a legendre transform of the lagrangian l l and vice versa. Inverse legendre transform in biochemical thermodynamics.
Solution of legendre equation legendre polynomials 3. These functions appear as solutions of boundary value problems in physics and engineering. If the function is convex and differentiable, then the supporting hyperplanes correspond to the derivative at each point, so the legendre transform is a reencoding of a functions information in terms of its derivative. As is to be expected, if the hamiltonian was generated by a legendre transformation from a lagrangian, then this lagrangian can be recovered. Use the legendre transformation rule to transform the enthalpy hs, p into a function of the variables s and v. Legendre transform is a special case of jacobi transform. But when experimental thermodynamic data involve an inconveniently large number of intensive variables, inverse legendre transforms are needed to reduce the number of independent intensive variables.
The legendre transform of the function in a hilbert space with scalar product is the function the legendre transformation, based on a change of variables, is a special case of a proximity transformation. In particular, we consider the symmetries associated with the inverse legendre transform, extreme values, and derivative relations. The energy and entropy representations we have noted that both su,v,nand. Legendre transforms in chemical thermodynamics chemical. Legendre transforms in chemical thermodynamics sciencedirect. Classical results are obtained according to the classical limit of the quantum mechanical expressions. The journal of chemical thermodynamics 2002, 34 11. Jun 28, 2017 the above equation holds for each pair satisfying, so we can explicitly confirm that 3 it is also clear from looking at a graph that a function inverse is symmetric i.
Introduction the legendre transform lt is a commonly used mathematical tool in upper division and graduate physics. Legendre transforms and their application in thermodynamics. The lagrangian function l is a function of coordinates and velocities, but it is often more convenient to define the hamiltonian function h with a legendre transform because the. Mckay3 1department of physics, virginia polytechnic institute and state university, blacksburg, va 24061 usa 2department of physics, university of maryland, college park, md 20742 usa and 3department of physics and astronomy, university of maine, orono, me, 04469 usa. The lagrangian function l is a function of coordinates and velocities, but it is often more convenient to define the hamiltonian function h with a legendre transform. The legendre transformation is discussed in detail in part iv, where it plays an important role in plasticity theory, and other topics. Legendre transform in twodimensional thermodynamics. In mathematics and physics, the legendre transformation, named after adrienmarie legendre, is an involutive transformation on the realvalued convex functions of one real variable. A graphical derivation of the legendre transform sam kennerly april 12, 2011 this work is licensed under the creative commons attribution 3. Show that this new function is the negative of the internal energy, and that the two differential relations correspond to two of the relations in table 4. In reality, the different ways to describe the system that differ by a legendre transformation are equally fundamental or equally natural so its often useful to be familiar with all of them and to know what. The legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and.