Liouville's Theorem

7 Key excerpts on "Liouville's Theorem"

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  Statistical Mechanics in a Nutshell

  • Luca Peliti(Author)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  which corresponds to Liouville's Theorem for an infinitesimal phase space region. The theorem can be extended to arbitrary regions by decomposing them into the union of infinitesimal ones. The fact that the canonical equations conserve the volume of regions in phase space is the special property of phase space we were looking for.

  Liouville's Theorem has an important corollary. Let ρ(x, t) be the local probability density of finding a system in the neighborhood of phase space point x. Given a small region dΓ of phase space around point x, the probability of finding the given system in dΓ at time t is given by

  We want to evaluate the way ρ(x, t) changes if the system evolves according to the canonical equations. Let us consider an arbitrary small region dΓ0 around x0 , and let ρ(x, 0) be the initial distribution at t = 0. Then, the probability dP(t) that the system is found at time t within the evolute dΓ

  t

  of Γ0 does not vary by definition, since if its initial condition was lying within dΓ0 at t = 0, its evolved point must lie within dΓ

  t

  at time t. We have, therefore,

  Thus,

  where in the last equality we have taken into account Liouville's Theorem. Therefore, the local probability density does not change if one moves along with the local velocity dictated by the canonical equations.

  Let us evaluate the expression of the local probability density change. We have

  Since Liouville's Theorem implies that dρ/dt = 0, we obtain

  The expression on the right-hand side is called the Poisson bracket of the Hamiltonian H with ρ, and is usually denoted by [,]PB . This equation is known as Liouville's equation.

  Let A(x) be a function defined on the phase space of the system, e.g., an observable. Let us evaluate the change of its instantaneous value as the system evolves according to the canonical equations of motion. We have

  and

  Thus, the derivative of A

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  An Introduction to the Physics of Particle Accelerators

  • Mario Conte, William W MacKay;;;(Authors)
  • 2008(Publication Date)
  • WSPC

    (Publisher)

  3 Mechanics of Trajectories In this chapter, the general theory of particle trajectories moving in a six dimen-sional phase space is examined in terms of generalized transformations and Hamil-tonian dynamics. For small deviations about a design trajectory, the linearized transformation yields a matrix which is the Jacobian matrix for the transforma-tion. Liouville’s theorem 1 , 2 , which requires the conservation of particle density in phase space for nondissipative systems, is shown to be equivalent to the requirement that the Jacobian matrix have a unit determinant. The trajectories are studied in terms of Hamiltonian dynamics, and the transformations are shown to be sym-plectic* for a canonical choice of spatial and momentum coordinates. Finally, the standard coordinates of Chapter 2 are obtained using the paraxial approximation, and their limitations are studied. 3.1 Liouville’s theorem For a large group of particles, a density function f ( x, y, z, p x , p y , p z , t ) may be defined for the number of particles per volume of six dimensional phase space with coordinates given by the three spatial coordinates x , y , and z , and their correspond-ing momentum coordinates p x , p y , and p z . We are assuming that the number of particles is large enough so we may treat f as a continuous function of the phase space coordinates and ignore the fact that the particles are really point objects. In relation to accelerator physics, Liouville’s theorem may be stated as follows: In the local region of a particle, the particle density in phase space is constant, provided that the particles move in a general field consisting of magnetic fields and of fields whose forces are independent of velocity. Here we are ignoring the effect of radiation due to the acceleration of the charges. Liouville’s theorem does not hold for systems with dissipation of energy, which is generally driven by velocity dependent forces.

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  Optics of Charged Particles

  • Hermann Wollnik(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Any attempt to reduce the size of this volume by electromagnetic fields is in vain, although it may often be desirable. 5.1 LIOUVILLE'S T H E O R E M A N D FIRST-ORDER TRANSFER MATRICES In a first-order approximation used up to now in Chapters 1, 3, and 4 and also here, the particle motions in the three space coordinates are independent of each other. This splits Liouville's Theorem into the conserva-tion of three two-dimensional phase-space areas at any time i f : dXi d p x i = const x , dyi dp yi = constx i dp zi = const z . At positions with vanishing vector potential, we read from Eqs. (2.22a) and (2.22b) p xi = a i n i p r and p yi = b ^ P r so that the first two of these statements become Pr J J dXitii d a t = p r J j dXiitid(sin a t ) = const x , (5.1a) Fig. 5.1 . The two-dimensional x, p x phase-space area of a beam of particles at (a) ζ = z x and (b) ζ = z 2 . Note the varied shapes and equal sizes of the indicated phase-space areas. 140 5 Charged Particle B e a m s and Phase S p a c e Pr dyn dbi = p, dy^d (sin ß t ) = C o n s t a (5.1b) Here, n, is the locally varying refractive index, and a, as well as 6, are taken from Eqs. (2.22a) and (2.22b) for the cases of planar motions, i.e., /3, = 0 for Eq. (5.1a) and a, = 0 for Eq. (5.1b). To accept Eq. (5.1a) as correct, remember that X, = (χ,, a,) of a particle trajectory at z, can be determined from X x = (x x , a x ) at z x by X, = T X X X , which reads explicitly, Remembering now the integration rules in a multidimensional space (see, for instance, Stromberg, 1981), we can rewrite Eq. (5.1a) as where G describes regions in the x, a space at z, and at z x . Because of |T X | = n x /n 2 as illustrated in Eq. (2.23a) and proved in Eq. (8.25a), Eq. (5.1a) is thus correct, as is Eq. (5.1b) analogously.

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  Dynamics of Particles and the Electromagnetic Field

  (With CD-ROM)

  • Slobodan Danko Bosanac(Author)
  • 2005(Publication Date)
  • WSPC

    (Publisher)

  Chapter 8 Lorentz Invariant Liouville Equation 8.1 General Theory Formulating the L-invariant Liouville equation needs some care, in the first place because the phase space density is no longer a function of only three variables, position, time and momentum. If the momentum enters then it is imperative that its fourth component also enters, and therefore the phase space density has the functional dependence pL(F, 24,@,,~4). The equation it satisfies, Liouville equation, is derived by applying the same arguments as in (1.10) except that time should be replaced by the invariant time, and therefore it is given by drx4&4~L + dr?. V~PL + dr~4dpq~L + 45. VppL = 0. (8.1) The essential difference from (1.11) is in the additional term that involves equation for the fourth component of the four-momentum. However, this additional term is obsolete because p4 satisfies the equation (6.25), and therefore it is not an independent variable. This fact is taken into account by parametrizing the phase space density as PL(F,24,$,P4) = p(Fix4i@?P4)6 (Pi -P 2 -m2c2) in which case Liouville equation (8.1) separates into two: one for the pos- itive value of p4 and the other for its negative value. When this para- metrization is taken into account then the equation that the function 141 142 Dynamics of Particles and the Electromagnetic Field ps (F, x4 , 6, s JjTiSZG) satisfies is where s is the sign of p4that was introduced in (7.4). Therefore there are two disjoint Liouville equations, one for each sign of the fourth component of the four-momentum, the meaning of which will be discussed later. In further analysis only the positive sign is assumed. Solution of the L-invariant Liouville equation (8.2) is obtained in essen- tially the same way as the solution for (1.11) except that the care should be taken that trajectories are solutions of equations (7.2) and (7.3). This point is not discussed in great details, instead few examples are analyzed.

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  Stochastic Dynamics and Boltzmann Hierarchy

  • Dmitri Ya. Petrina(Author)
  • 2009(Publication Date)
  • De Gruyter

    (Publisher)

  If ij .p i NUL p j / 0, then one chooses x as the initial point. These boundary conditions of collisions at the initial time t D 0 play an important role in the Liouville equation. It is clear that these boundary conditions of collisions, when momenta have jumps, are also satisfied at arbitrary time t ¤ 0. Namely, if Q i .t/ NUL Q j .t/ D a; j t j ¤ 0, 18 1 System of hard spheres then .P i .t/; P j .t// ! .P i .t/; P j .t// for NUL P i .t/ NUL P j .t/ 0 and t > 0, or for NUL P i .t/ NUL P j .t/ 0 and t < 0. We summarize the properties of trajectories described above in the following theo-rem: Theorem 1.1. For a system of N hard spheres, a unique trajectory X.t; x/ exists for almost all initial data outside certain hypersurfaces of lower dimension. The trajec-tory X.t; x/ is continuously differentiable on intervals of time between collisions with respect to time t and initial data x and has the group property (1.2.10). The trajectory satisfies the above-described boundary conditions of collisions. It is obvious that the trajectory X.t; x/ belongs to the admissible configuration, i.e., to the completion of the forbidden configuration. 1.2.3 Liouville theorem We define a mapping T t (a flow) of the phase space onto itself as a shift along the trajectory X.t; x/ . Namely, T t x D X.t; x/; t 2 R 1 ; x 2 R 3 N R 3 N ; for all x for which the trajectory X.t; x/ exists. This means that the mapping T t is defined for almost all x , i.e., outside certain hypersurfaces. On time intervals between collisions, the Jacobian @X.t;x/ @x of the mapping T t is equal to one. It is easy to check that the transformation of momenta at the instant of collision (1.2.3) also has the Jaco-bian @.P i .t/;P j .t// @.P i .t/;P j .t// equal to one. One can calculate the Jacobian @X.t;x/ @x consequently on the intervals between colli-sions, where it is equal to one, and at the instants of collisions, where it is again equal to one. The obtained results are formulated as the Liouville theorem.

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  Particle Physics - Vi Jorge Andre Swieca Summer School

  Proceedings of the VI Jorge André Swieca Summer School

  • M O C Gomes, Oscar J P Eboli, A Santoro(Authors)
  • 1992(Publication Date)
  • World Scientific

    (Publisher)

  We further showed in §111.D that to every solution t/> of the free field equations with p^ > 0, there corresponds a unique Liouville solution and vice versa. The restriction on momenta comes about essentially because the collective mode if> 0 -f p^t of the free case contains twice as many states as the Liouville collective mode. The Liouville collective mode effectively evolves in a quantum mechanical Liouville potential, and every incoming wave is also automatically outgoing. Clearly, these classical statements directly translate into a result on the corre-spondence between the spectra. The easiest way to see this is to decompose the Liouville Hamiltonian as follows: (6.A.3) where h is the Hamiltonian of the collective mode only h = p 2 + v 2 e 2a +° + !j (6.A.4)

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  Statistical Mechanics and Applications in Condensed Matter

  • Carlo Di Castro, Roberto Raimondi(Authors)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  Representative points in  cannot be created or destroyed as a consequence of the Liouville theorem, so that during its evolution ρ must satisfy the continuity equation ∂ρ ∂ t + ∇ · ρ v = ∂ρ ∂ t + ∇ρ · v + ρ ∇ · v = 0. (3.10) Since by the Hamilton equations ∇ · v ≡ 0 (see Problem 3.1), we can conclude that the total time derivative of ρ is zero, dρ dt ≡ ∂ρ ∂ t + ∇ρ · ˙ x = 0. For an equilibrium ensemble, the distribution function cannot depend explicitly on time and then must depend on the  space coordinates only through the Hamiltonian. It is clear that the distribution f (q, p, t ) in the six-dimensional μ-space corresponds to a coarse-grained description of the system: given a point in  space, f (q, p, t ) is determined, and vice versa, given f there are several micro-states compatible with the same f (q, p, t ). Hence to a given f corresponds a volume in . Again following Boltzmann we can discretize the distribution function by specifying the set of M occupation numbers n i (i = 1, . . . , M ) of M cells of volume ω i with the condition that M  i =1 n i = N , M  i =1  i n i = E ,  i = p 2 i 2m . (3.11) In this way the ratio n i /ω i represents the discretized value of the distribution function in the volume element ω i of dimension of an action. The volume in  corresponding to a given set of occupation numbers, {n i }, is V  ({n i }) = N !  i n i !  i ω n i i , (3.12) where the combinatorial factor N !/  i n i ! comes from the distribution of N distinguishable objects in the M boxes, the i -th box containing n i objects. We want to determine the set { ¯ n i } of occupation numbers that makes V  ({n i }) largest with the constraints (3.11). To this end it is convenient to study the maximum of the logarithm of the volume V  ({n i }): ln V  ({n i }) ∼ N ln N − N +  i n i ln ω i −  i (n i ln n i − n i ), = N ln N +  i n i ln ω i n i , (3.13)

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