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| {{About|Liouville's theorem in Hamiltonian mechanics||Liouville's theorem (disambiguation)}}
| | Epoxy resins are popular reactive intermediates utilized to produce a versatile group of thermosetting polymers. The term epoxy has been largely adapted for innumerable applications beyond fiber reinforced polymer composites. In the current market scene, these adhesives are bought and sold in most local hardware shops and are utilized as coatings for floors or binder in counter tops. Here is more info about [http://penae5.soup.io/post/430082517/Sobre-La-Versatilidad-De-La-Resina-Epoxi epoxi resina] check out our own web site. The gamut uses of these chemical compounds prevail to expand and variants of epoxies are consistently being developed to fit the requirements of industries and products they are utilized in.<br><br>These adhesives are commonly utilized in:<br><br>Industrial coatings<br><br>Non-skid coatings<br><br>Rigid foams<br><br>As a binder in mortars and cements<br><br>Fiber reinforced plastics<br><br>Encapsulating media and potting<br><br>Epoxy based resin is a material, which is manufactured by transforming liquid-polyether into infusible solids with the aid of apposite curing processes. Resins are usually produced by means of chemical reactions of bisphenol-A, epichlorohydrin and a few other chemical reactants. The availability and trading or these chemical compounds have resulted in higher economic turnover across the globe.<br><br>The epoxies are well-known for their excellent chemical and mechanical properties including electrical insulating properties, electrical and chemical resistance properties and adhesive properties. They are available in the form of diluent, liquids, solids, special and multi-functional resins, Bisphenol and cycloaliphatic forms.<br><br>Epoxy resins have found an eminent place of an important constituent in myriad applications and uses.<br><br>It is largely proffered as coatings owing to its resistance properties against metals and alkali. Having such properties has paved its way to be utilized in floor coatings and paints in industrial and automobile industry. High color retention, resistant to electrical insulation and heat are a few major qualities of thee resins.<br><br>This chemical compound is used for tens of thousands of purposes including professional as well as domestic applications. They are widely used in the industries specializing in dentistry, fiber optics, opto-electronics and numerous other industrial applications. When it comes to domestic applications, these resins are predominantly utilized in manufacturing of glassware, leather and wooden products. These high-utility resins are also used by goldsmiths to render shape to the handcrafted pieces of ornaments like earrings, bracelets and necklaces etc.<br><br>These adhesives are also referred to as engineering adhesives and are commonly utilized in the sports, automobile and marine industry. An ideal bonding for stone, plastics, glass, metal and wood, this adhesive has brilliant chemical and heat resistance. Other spheres of its usages and utilities comprises of equipment and tool production; in construction engineering as construction adhesives; in the electronic industry for switchgear, insulators, transformers, bushings and motors and in the construction industry as the preservatives for floors, airport runways, roads, high-pressure moldings and concrete floors.<br><br>Epoxy resins boast of infinite important uses and are largely utilized in several different industries throughout the world. Easily available, this product is extremely durable and affordable, which in turns helps in the economic growth of a business. |
| In [[physics]], '''Liouville's theorem''', named after the French mathematician [[Joseph Liouville]], is a key theorem in classical [[statistical mechanics|statistical]] and [[Hamiltonian mechanics]]. It asserts that the [[phase space|phase-space]] distribution function is constant along the trajectories of the system — that is that the density of system points in the vicinity of a given system point travelling through phase-space is constant with time. | |
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| There are also related mathematical results in [[symplectic topology]] and [[ergodic theory]].
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| ==Liouville equations==
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| [[File:Hamiltonian flow classical.gif|frame|Evolution of an ensemble of [[Hamiltonian mechanics|classical]] systems in [[phase space]] (top). Each system consists of one massive particle in a one-dimensional [[potential well]] (red curve, lower figure). Whereas the motion of an individual member of the ensemble is given by [[Hamilton's equation]]s, Lioville's equations describe the flow of whole. The motion is analogous to a dye in an incompressible fluid.]] | |
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| These Liouville equations describe the time evolution of the phase space [[distribution function]]. Although the equation is usually referred to as the "Liouville equation", this equation was in fact first published by [[Josiah Willard Gibbs]] in 1902.<ref>Page 9, {{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=[[Elementary Principles in Statistical Mechanics]] |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York}}</ref> It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838.<ref>[J. Liouville, Journ. de Math., 3, 349(1838)].</ref>
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| Consider a [[Hamiltonian system|Hamiltonian dynamical system]] with [[canonical coordinates]] <math>q_i</math> and [[conjugate momenta]] <math>p_i</math>, where <math>i=1,\dots,n</math>. Then the phase space distribution <math>\rho(p,q)</math> determines the probability <math>\rho(p,q)\,d^nq\,d^n p</math> that the system will be found in the infinitesimal phase space volume <math>d^nq\,d^n p</math>. The '''Liouville equation''' governs the evolution of <math>\rho(p,q;t)</math> in time <math>t</math>:
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| :<math>\frac{d\rho}{dt}=
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| \frac{\partial\rho}{\partial t}
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| +\sum_{i=1}^n\left(\frac{\partial\rho}{\partial q_i}\dot{q}_i
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| +\frac{\partial\rho}{\partial p_i}\dot{p}_i\right)=0.</math>
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| Time derivatives are denoted by dots, and are evaluated according to [[Hamilton's equations]] for the system. This equation demonstrates the conservation of density in phase space (which was [[Willard Gibbs|Gibbs]]'s name for the theorem). Liouville's theorem states that
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| :''The distribution function is constant along any trajectory in phase space.''
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| A simple [https://en.wikiversity.org/w/index.php?title=Topic:Advanced_Classical_Mechanics/Phase_Space&oldid=1135602| proof of the theorem] is to observe that the evolution of <math>\rho</math> is ''defined'' by the [[continuity equation]]:
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| :<math>\frac{\partial\rho}{\partial t}+\sum_{i=1}^n\left(\frac{\partial(\rho\dot{q}_i)}{\partial q_i}+\frac{\partial(\rho\dot{p}_i)}{\partial p_i}\right)=0.</math>
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| That is, the tuplet <math>(\rho, \rho\dot{q}_i,\rho\dot{p}_i)</math> is a [[conserved current]]. Notice that the difference between this and Liouville's equation are the terms
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| :<math>\rho\sum_{i=1}^n\left(
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| \frac{\partial\dot{q}_i}{\partial q_i}
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| +\frac{\partial\dot{p}_i}{\partial p_i}\right)
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| =\rho\sum_{i=1}^n\left(
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| \frac{\partial^2 H}{\partial q_i\,\partial p_i}
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| -\frac{\partial^2 H}{\partial p_i \partial q_i}\right)=0,</math> | |
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| where <math>H</math> is the Hamiltonian, and Hamilton's equations have been used. That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem that the [[convective derivative]] of the density, <math>d \rho/dt</math>, is zero follows from the equation of continuity by noting that the 'velocity field' <math>(\dot p , \dot q)</math> in phase space has zero divergence (which follows from Hamilton's relations).
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| Another illustration is to consider the trajectory of a cloud of points through phase space. It is straightforward to show that as the cloud stretches in one coordinate – <math>p_i</math> say – it shrinks in the corresponding <math>q^i </math> direction so that the product <math>\Delta p_i \, \Delta q^i </math> remains constant.
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| Equivalently, the existence of a conserved current implies, via [[Noether's theorem]], the existence of a [[symmetry]]. The symmetry is invariant under time translations, and the [[generator (mathematics)|generator]] (or [[Noether charge]]) of the symmetry is the Hamiltonian.
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| ==Other formulations==
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| === Poisson bracket ===
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| The theorem is often restated in terms of the [[Poisson bracket]] as
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| :<math>\frac{\partial\rho}{\partial t}=-\{\,\rho,H\,\}</math>
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| or in terms of the '''Liouville operator''' or '''Liouvillian''',
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| :<math>\mathrm{i}\hat{\mathbf{L}}=\sum_{i=1}^{n}\left[\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{\partial }{\partial p_{i}}\right]=\{\cdot,H\}</math>
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| as
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| :<math>\frac{\partial \rho }{\partial t}+{\mathrm{i}\hat{\mathbf{L}}}\rho =0.</math>
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| === Ergodic theory ===
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| In [[ergodic theory]] and [[dynamical systems]], motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. In [[Hamiltonian mechanics]], the phase space is a [[differentiable manifold|smooth manifold]] that comes naturally equipped with a smooth [[Measure (mathematics)|measure]] (locally, this measure is the 6''n''-dimensional [[Lebesgue measure]]). The theorem says this smooth measure is invariant under the [[Hamiltonian flow]]. More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow. The Hamiltonian case then becomes a corollary.
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| === Symplectic geometry ===
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| In terms of [[symplectic geometry]], the phase space is represented as a [[symplectic manifold]]. The theorem then states that the natural [[volume form]] on a symplectic manifold is invariant under the Hamiltonian flows. The symplectic structure is represented as a [[2-form]], given as a sum of [[wedge product]]s of d''p''<sub>''i''</sub> with d''q''<sup>i</sup>. The volume form is the top [[exterior power]] of the symplectic 2-form, and is just another representation of the measure on the phase space described above. One formulation of the theorem states that the [[Lie derivative]] of this volume form is zero along every Hamiltonian vector field.
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| In fact, the symplectic structure itself is preserved, not only its top exterior power. For this reason, in this context, the symplectic structure is also called Poincaré invariant. Hence the theorem about Poincaré invariant is a generalization of the Liouville's theorem.
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| ===Quantum Liouville equation===
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| The analog of Liouville equation in [[quantum mechanics]] describes the time evolution of a [[Density matrix|mixed state]]. [[Canonical quantization]] yields a quantum-mechanical version of this theorem, the [[Von Neumann equation]]. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by [[commutator]]s. In this case, the resulting equation is<ref>[http://books.google.com/books?id=0Yx5VzaMYm8C&pg=PA110 ''The theory of open quantum systems'', by Breuer and Petruccione, p110].</ref><ref>[http://books.google.com/books?id=o-HyHvRZ4VcC&pg=PA16 ''Statistical mechanics'', by Schwabl, p16].</ref>
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| :<math>\frac{\partial \rho}{\partial t}=\frac{1}{i \hbar}[H,\rho]</math>
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| where ρ is the [[density matrix]].
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| When applied to the [[expectation value]] of an [[observable]], the corresponding equation is given by [[Ehrenfest's theorem]], and takes the form
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| :<math>\frac{d}{dt}\langle A\rangle = \frac{1}{i \hbar}\langle [A,H] \rangle</math>
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| where <math>A</math> is an observable. Note the sign difference, which follows from the assumption that the operator is stationary and the state is time-dependent.
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| ==Remarks==
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| *The Liouville equation is valid for both equilibrium and nonequilibrium systems. It is a fundamental equation of [[non-equilibrium statistical mechanics]].
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| *The Liouville equation is integral to the proof of the [[fluctuation theorem]] from which the [[second law of thermodynamics]] can be derived. It is also the key component of the derivation of [[Green-Kubo relations]] for linear transport coefficients such as shear [[viscosity]], [[thermal conductivity]] or [[electrical conductivity]].
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| * Virtually any textbook on [[Hamiltonian mechanics]], advanced [[statistical mechanics]], or [[symplectic geometry]] will derive<ref>[for a particularly clear derivation see "The Principles of Statistical Mechanics" by R.C. Tolman , Dover(1979), p48-51].</ref> the Liouville theorem<ref>http://hepweb.ucsd.edu/ph110b/110b_notes/node93.html Nearly identical to proof in this Wikipedia article. Assumes (without proof) the n-dimensional continuity equation. Retrieved 01/06/2014. </ref><ref>http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_2/node2.html A rigorous proof based on how the Jacobian volume element transforms under Hamiltonian mechanics. Retrieved 01/06/2014. </ref><ref>http://www.pma.caltech.edu/~mcc/Ph127/a/Lecture_3.pdf Uses the n-dimensional divergence theorem (without proof) Retrieved 01/06/2014. </ref>
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| ==See also==
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| * [[Reversible reference system propagation algorithm]] (r-RESPA)
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| ==References==
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| * ''Modern Physics'', by R. Murugeshan, S. Chand publications
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| * Liouville's theorem in curved space-time : ''Gravitation'' § 22.6, by Misner,Thorne and Wheeler, Freeman
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| {{reflist}}
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| [[Category:Hamiltonian mechanics]]
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| [[Category:Theorems in dynamical systems]]
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| [[Category:Statistical mechanics theorems]]
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Epoxy resins are popular reactive intermediates utilized to produce a versatile group of thermosetting polymers. The term epoxy has been largely adapted for innumerable applications beyond fiber reinforced polymer composites. In the current market scene, these adhesives are bought and sold in most local hardware shops and are utilized as coatings for floors or binder in counter tops. Here is more info about epoxi resina check out our own web site. The gamut uses of these chemical compounds prevail to expand and variants of epoxies are consistently being developed to fit the requirements of industries and products they are utilized in.
These adhesives are commonly utilized in:
Industrial coatings
Non-skid coatings
Rigid foams
As a binder in mortars and cements
Fiber reinforced plastics
Encapsulating media and potting
Epoxy based resin is a material, which is manufactured by transforming liquid-polyether into infusible solids with the aid of apposite curing processes. Resins are usually produced by means of chemical reactions of bisphenol-A, epichlorohydrin and a few other chemical reactants. The availability and trading or these chemical compounds have resulted in higher economic turnover across the globe.
The epoxies are well-known for their excellent chemical and mechanical properties including electrical insulating properties, electrical and chemical resistance properties and adhesive properties. They are available in the form of diluent, liquids, solids, special and multi-functional resins, Bisphenol and cycloaliphatic forms.
Epoxy resins have found an eminent place of an important constituent in myriad applications and uses.
It is largely proffered as coatings owing to its resistance properties against metals and alkali. Having such properties has paved its way to be utilized in floor coatings and paints in industrial and automobile industry. High color retention, resistant to electrical insulation and heat are a few major qualities of thee resins.
This chemical compound is used for tens of thousands of purposes including professional as well as domestic applications. They are widely used in the industries specializing in dentistry, fiber optics, opto-electronics and numerous other industrial applications. When it comes to domestic applications, these resins are predominantly utilized in manufacturing of glassware, leather and wooden products. These high-utility resins are also used by goldsmiths to render shape to the handcrafted pieces of ornaments like earrings, bracelets and necklaces etc.
These adhesives are also referred to as engineering adhesives and are commonly utilized in the sports, automobile and marine industry. An ideal bonding for stone, plastics, glass, metal and wood, this adhesive has brilliant chemical and heat resistance. Other spheres of its usages and utilities comprises of equipment and tool production; in construction engineering as construction adhesives; in the electronic industry for switchgear, insulators, transformers, bushings and motors and in the construction industry as the preservatives for floors, airport runways, roads, high-pressure moldings and concrete floors.
Epoxy resins boast of infinite important uses and are largely utilized in several different industries throughout the world. Easily available, this product is extremely durable and affordable, which in turns helps in the economic growth of a business.