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| {{About|statistical mechanics|other uses|partition function (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. 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. If you loved this article so you would like to collect more info about [http://prattz1.blog.com/2014/05/28/la-fibra-de-vidrio-un-elemento-provechoso-con-considerable-capacidad-para-el-reciclaje/ resina epoxi] please visit our web-page. 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. |
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| {{statistical mechanics}}
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| In [[physics]], a '''partition function''' describes the [[statistics|statistical]] properties of a system in [[thermodynamic equilibrium]]. They are [[function (mathematics)|functions]] of [[temperature]] and other parameters, such as the [[volume]] enclosing a gas. Most of the aggregate [[thermodynamics|thermodynamic]] variables of the system, such as the [[energy|total energy]], [[Thermodynamic free energy|free energy]], [[entropy]], and [[pressure]], can be expressed in terms of the partition function or its [[derivative]]s.
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| There are actually several different types of partition functions, each corresponding to different types of [[statistical ensemble]] (or, equivalently, different types of [[Thermodynamic free energy|free energy]].) The '''canonical partition function''' applies to a [[canonical ensemble]], in which the system is allowed to exchange [[heat]] with the environment at fixed temperature, volume, and [[number of particles]]. The '''grand canonical partition function''' applies to a [[grand canonical ensemble]], in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and [[chemical potential]]. Other types of partition functions can be defined for different circumstances; see [[partition function (mathematics)]] for generalizations.
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| == Canonical partition function ==
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| === Definition ===
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| As a beginning assumption, assume that a thermodynamically large system is in thermal contact with the environment, with a temperature ''T'', and both the volume of the system and the number of constituent particles are fixed. This kind of system is called a [[canonical ensemble]]. Let us label with ''s'' = 1, 2, 3, ... the ''exact'' states ([[Microstate (statistical mechanics)|microstates]]) that the system can occupy, and denote the total energy of the system when it is in microstate ''s'' as ''E<sub>s</sub>''. Generally, these microstates can be regarded as analogous to discrete [[quantum state]]s of the system.
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| The '''canonical partition function''' is
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| : <math> Z = \sum_{s} \mathrm{e}^{- \beta E_s}</math> , | |
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| where the "inverse temperature", ''[[Thermodynamic beta|β]]'', is conventionally defined as
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| : <math>\beta \equiv \frac{1}{k_BT}</math>
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| with ''k''<sub>B</sub> denoting [[Boltzmann's constant]]. The [[Exponential function|exponential]] factor exp(−''βE<sub>s</sub>'') is known as the [[Boltzmann factor]]. (For a detailed derivation of this result, see [[canonical ensemble]]). In systems with multiple quantum states ''s'' sharing the same ''E<sub>s</sub>'', it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by ''j'' ) as follows:
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| : <math> Z = \sum_{j} g_j \cdot \mathrm{e}^{- \beta E_j}</math>,
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| where ''g''<sub>j</sub> is the degeneracy factor, or number of quantum states ''s'' which have the same energy level defined by ''E<sub>j</sub>'' = ''E<sub>s</sub>''.
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| The above treatment applies to ''quantum'' [[statistical mechanics]], where a physical system inside a finite-sized box will typically have a discrete set of [[energy]] [[eigenstates]], which we can use as the states ''s'' above. In ''classical'' statistical mechanics, it is not really correct to express the partition function as a sum of discrete terms, as we have done. In [[classical mechanics]], the position and momentum variables of a particle can vary continuously, so the set of microstates is actually [[uncountable set|uncountable]]. In this case we must describe the partition function using an [[integral]] rather than a sum. For instance, the partition function of a gas of ''N'' identical classical particles is
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| : <math>Z=\frac{1}{N! h^{3N}} \int \, \exp[-\beta H(p_1 \cdots p_N, x_1
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| \cdots x_N)] \; d^3p_1 \cdots d^3p_N \, d^3x_1 \cdots d^3x_N </math>
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| where
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| :''p<sub>i</sub>'' indicate particle momenta
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| :''x<sub>i</sub>'' indicate particle positions
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| :''d''<sup>3</sup> is a shorthand notation serving as a reminder that the ''p<sub>i</sub>'' and ''x<sub>i</sub>'' are vectors in three dimensional space, and
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| :''H'' is the classical [[Hamiltonian mechanics|Hamiltonian]].
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| The reason for the [[factorial]] factor ''N''! is discussed [[#Partition functions of subsystems|below]]. For simplicity, we will use the discrete form of the partition function in this article. Our results will apply equally well to the continuous form. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not [[dimensionless]]. To make it into a dimensionless quantity, we must divide it by ''h''<sup>3''N''</sup> where ''h'' is some quantity with units of [[action (physics)|action]] (usually taken to be [[Planck's constant]]). | |
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| In quantum mechanics, the partition function can be more formally written as a [[trace (linear algebra)|trace]] over the [[mathematical formulation of quantum mechanics|state space]] (which is independent of the choice of [[basis (linear algebra)|basis]]):
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| : <math>Z = \operatorname{tr} ( \mathrm{e} ^{-\beta\hat{H}} )</math> ,
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| where ''Ĥ'' is the [[Hamiltonian (quantum mechanics)|quantum Hamiltonian operator]]. The exponential of an operator can be defined using the [[Characterizations of the exponential function|exponential power series]].
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| The classical form of ''Z'' is recovered when the trace is expressed in terms
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| of [[coherent state]]s
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| <ref>J. R. Klauder, B.-S. Skagerstam,
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| ''Coherent States --- Applications in Physics and Mathematical Physics'',
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| World Scientific, 1985, p. 71-73.</ref>
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| and when quantum-mechanical [[uncertainty principle|uncertainties]] in the position and momentum of a particle | |
| are regarded as negligible. Formally, one inserts under the trace for each degree of freedom the identity: | |
| :<math>
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| \boldsymbol{1} = \int |x,p\rangle\,\langle x,p|~\frac{ dx\, dp}{h}
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| </math>
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| where {{!}}''x'', ''p''{{rangle}} is a normalised Gaussian wavepacket centered at
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| position ''x'' and momentum ''p''. Thus,
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| :<math>
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| Z = \int \operatorname{tr} \left( \mathrm{e}^{-\beta\hat{H}} |x,p\rangle\,\langle x,p| \right)
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| \frac{ dx\, dp}{h}
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| = \int\langle x,p| \mathrm{e} ^{-\beta\hat{H}}|x,p\rangle ~\frac{ dx\, dp}{h}
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| </math> | |
| A coherent state is an approximate eigenstate
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| of both operators <math> \hat{x} </math> and <math> \hat{p} </math>, | |
| hence also of the Hamiltonian ''Ĥ'', with errors of the size of
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| the uncertainties. If Δ''x'' and Δ''p'' can be regarded as
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| zero, the action of ''Ĥ'' reduces to multiplication by the classical
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| Hamiltonian, and ''Z'' reduces to the classical configuration integral.
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| === Meaning and significance ===
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| It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, let us consider what goes into it. The partition function is a function of the temperature ''T'' and the microstate energies ''E''<sub>1</sub>, ''E''<sub>2</sub>, ''E''<sub>3</sub>, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.
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| The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability ''P<sub>s</sub>'' that the system occupies microstate ''s'' is
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| : <math>P_s = \frac{1}{Z} \mathrm{e}^{- \beta E_s}. </math>
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| The partition function thus plays the role of a normalizing constant (note that it does ''not'' depend on ''s''), ensuring that the probabilities sum up to one:
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| : <math>\sum_s P_s = \frac{1}{Z} \sum_s \mathrm{e}^{- \beta E_s} = \frac{1}{Z} Z
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| = 1. </math>
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| This is the reason for calling ''Z'' the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. The letter ''Z'' stands for the [[German language|German]] word ''Zustandssumme'', "sum over states". This notation also implies another important meaning of the partition function of a system: it counts the (weighted) number of states a system can occupy. Hence if all states are equally probable (equal energies) the partition function is the total number of possible states. Often this is the practical importance of Z.
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| === Calculating the thermodynamic total energy ===
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| In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the [[expected value]], or [[ensemble average]] for the energy, which is the sum of the microstate energies weighted by their probabilities:
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| : <math>\langle E \rangle = \sum_s E_s P_s = \frac{1}{Z} \sum_s E_s
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| e^{- \beta E_s} = - \frac{1}{Z} \frac{\partial}{\partial \beta}
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| Z(\beta, E_1, E_2, \cdots) = - \frac{\partial \ln Z}{\partial \beta}
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| </math> | |
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| or, equivalently,
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| : <math>\langle E\rangle = k_B T^2 \frac{\partial \ln Z}{\partial T}.</math>
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| Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner
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| : <math>E_s = E_s^{(0)} + \lambda A_s \qquad \mbox{for all}\; s </math>
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| then the expected value of ''A'' is
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| : <math>\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta}
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| \frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math>
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| This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set ''λ'' to zero in the final expression. This is analogous to the [[source field]] method used in the [[path integral formulation]] of [[quantum field theory]].
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| === Relation to thermodynamic variables ===
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| In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.
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| As we have already seen, the thermodynamic energy is
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| : <math>\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math>
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| The [[variance]] in the energy (or "energy fluctuation") is
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| : <math>\langle (\Delta E)^2 \rangle \equiv \langle (E - \langle
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| E\rangle)^2 \rangle = \frac{\partial^2 \ln Z}{\partial \beta^2}.</math>
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| The [[heat capacity]] is
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| : <math>C_v = \frac{\partial \langle E\rangle}{\partial T} = \frac{1}{k_B T^2} \langle (\Delta E)^2 \rangle.</math>
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| The [[entropy]] is
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| : <math>S \equiv -k_B\sum_s P_s\ln P_s= k_B (\ln Z + \beta \langle E\rangle)=\frac{\partial}{\partial T}(k_B T \ln Z) =-\frac{\partial A}{\partial T}</math>
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| where ''A'' is the [[Helmholtz free energy]] defined as ''A'' = ''U'' − ''TS'', where ''U'' = {{langle}}''E''{{rangle}} is the total energy and ''S'' is the [[entropy]], so that
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| : <math>A = \langle E\rangle -TS= - k_B T \ln Z.</math>
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| === Partition functions of subsystems ===
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| Suppose a system is subdivided into ''N'' sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ''ζ''<sub>1</sub>, ''ζ''<sub>2</sub>, ..., ''ζ''<sub>N</sub>, then the partition function of the entire system is the ''product'' of the individual partition functions:
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| : <math>Z =\prod_{j=1}^{N} \zeta_j.</math>
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| If the sub-systems have the same physical properties, then their partition functions are equal, ζ<sub>1</sub> = ζ<sub>2</sub> = ... = ζ, in which case
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| : <math>Z = \zeta^N.</math>
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| However, there is a well-known exception to this rule. If the sub-systems are actually [[identical particles]], in the [[quantum mechanics|quantum mechanical]] sense that they are impossible to distinguish even in principle, the total partition function must be divided by a ''N''! (''N'' [[factorial]]):
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| : <math>Z = \frac{\zeta^N}{N!}.</math>
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| This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the [[Gibbs paradox]].
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| ==Grand canonical partition function==
| |
| {{main|Grand canonical ensemble}}
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| We can define a '''grand canonical partition function''' for a [[grand canonical ensemble]], which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir.
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| The reservoir has a constant temperature ''T'', and a [[chemical potential]] ''μ''.
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| The grand canonical partition function, denoted by <math>\mathcal{Z}</math>, is the following sum over [[microstate (statistical mechanics)|microstates]]
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| :<math> \mathcal{Z}(\mu, V, T) = \sum_{i} \exp((N_i\mu - E_i)/k_B T). </math>
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| Here, each microstate is labelled by <math>i</math>, and has total particle number <math>N_i</math> and total energy <math>E_i</math>.
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| This partition function is closely related to the [[Grand potential]], <math>\Phi_{\rm G}</math>, by the relation
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| :<math> -k_B T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. </math>
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| This can be contrasted to the canonical partition function above, which is related instead to the [[Helmholtz free energy]].
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| It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble,
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| since here we consider not only variations in energy but also in particle number.
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| Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state <math>i</math>:
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| :<math> p_i = \frac{1}{\mathcal Z} \exp((N_i\mu - E_i)/k_B T) .</math>
| |
| | |
| An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas ([[Fermi-Dirac statistics]] for fermions, [[Bose-Einstein statistics]] for bosons), however it is much more generally applicable than that.
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| The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases.
| |
| | |
| The grand partition function is sometimes written (equivalently) in terms of alternate variables as<ref>{{cite isbn|9780120831807}}</ref>
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| :<math> \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), </math>
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| where <math>z \equiv \exp(\mu/kT)</math> is known as the ''activity'' or ''fugacity'' and <math>Z(N_i, V, T)</math> is the canonical partition function.
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| ==See also==
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| * [[Partition function (mathematics)]]
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| * [[Virial theorem]]
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| * [[Widom insertion method]]
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| ==References==
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| <references />
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| * Huang, Kerson, "Statistical Mechanics", John Wiley & Sons, New York, 1967.
| |
| * A. Isihara, "Statistical Physics", Academic Press, New York, 1971.
| |
| * Kelly, James J, [http://www.physics.umd.edu/courses/Phys603/kelly/Notes/IdealQuantumGases.pdf (Lecture notes)]
| |
| * L. D. Landau and E. M. Lifshitz, "Statistical Physics, 3rd Edition Part 1", Butterworth-Heinemann, Oxford, 1996.
| |
| * Vu-Quoc, L., [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral], 2008
| |
| {{Statistical mechanics topics}}
| |
| | |
| [[Category:Concepts in physics]]
| |
| [[Category:Partition functions| ]]
| |
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. 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. If you loved this article so you would like to collect more info about resina epoxi please visit our web-page. 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.