formulasearchengine - User contributions [en]

Matrix Chernoff bound

2014-02-05T05:10:39Z

68.65.166.81: Updating reference to improved Matrix Hoeffding inequality.

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Reproducing kernel Hilbert space

2014-02-03T05:33:59Z

68.65.164.57:

[[Image:Undulator.png|300px|thumb|Working of the undulator. 1: magnets, 2: electron beam entering from the upper left, 3: synchrotron radiation exiting to the lower right]]
[[Image:Aust.-Synchrotron,-Undulator,-14.06.2007.jpg|thumb|right|250px|A multipole wiggler, as used in the [[storage ring]] at the [[Australian Synchrotron]] to generate [[synchrotron radiation]]]]
An '''undulator''' is an [[insertion device]] from high-energy physics and usually part of a larger
installation, a [[synchrotron]] [[storage ring]]. It consists of a periodic structure of [[dipole magnet]]s. The static [[magnetic field]] is alternating along the length of the '''undulator''' with a wavelength <math>\lambda_u</math>. Electrons traversing the periodic magnet structure are forced to undergo oscillations and thus to radiate energy. The radiation produced in an undulator is very intense and concentrated in narrow energy bands in the spectrum. It is also [[Collimated_light|collimated]] on the orbit plane of the electrons. This radiation is guided through [[beamlines]] for experiments in various scientific areas.

The important dimensionless parameter

<math>K=\frac{e B \lambda_u}{2 \pi \beta m_e c}</math>{{cn|date=October 2013}}

where ''e'' is the electron charge, ''B'' is the magnetic field, ''<math>\beta</math>'' is the speed of the electron relative to the speed of light ''<math>(\beta=v/c)</math>'', ''<math>m_{e}</math>'' is the electron rest
mass, and ''c'' is the speed of light, characterizes the nature of the electron motion. For
<math>K\ll1</math> the oscillation amplitude of the motion is small and the radiation displays interference patterns which lead to narrow energy bands. If <math>K\gg1</math> the oscillation amplitude is bigger and the radiation contributions from each field period sum up independently, leading to a broad energy spectrum. In this regime of fields the device is no longer called an ''undulator''; it is called a [[wiggler (synchrotron)|wiggler]].

The usual description of the undulator is relativistic but classical. This means that though the precision calculation is tedious the undulator can be seen as a [[black box]]. An electron enters this box and an electromagnetic pulse exits through a small exit slit. The slit should be small enough such that only the main cone passes, so that the side lobes may be ignored.

Undulators can provide several orders of magnitude higher flux than a simple bending magnet and as such are in high demand at synchrotron radiation facilities. For an undulator with N periods, the [[Synchrotron_light_source#Brilliance|brightness]] can be up to <math>N^{2}</math> more than a bending magnet. The first factor of N occurs because the intensity is enhanced up to a factor of N at harmonic wavelengths due to the constructive interference of the fields emitted during the N radiation periods. The usual pulse is a sine with some envelope. The second factor of N comes from the reduction of the emission angle associated with these harmonics, which is reduced as 1/N. When the electrons come with half the period, they interfere destructively, the undulator stays dark. The same is true, if they come as a bead chain.

The polarization of the emitted radiation can be controlled by using permanent magnets to induce different periodic electron trajectories through the undulator. If the oscillations are confined to a plane the radiation will be linearly polarized. If the oscillation trajectory is helical, the radiation will be circularly polarized, with the handedness determined by the helix.

If the electrons follow the [[Poisson distribution]] a partial interference leads to a linear increase in intensity.
In the [[free electron laser]]<ref>Paolo Luchini, Hans Motz, ''Undulators and Free-electron Lasers'', Oxford University Press, 1990.</ref> the intensity increases exponentially with the number of electrons.

An undulator's [[figure of merit]] is [[spectral radiance]].

==History==
The first undulator was built by [[Hans Motz]] and his coworkers at [[Stanford]]<ref>{{Cite doi|10.1063.2F1.1700002}}</ref><ref>{{Cite doi|10.1063/1.1721389}}</ref> in 1952. It produced the first manmade coherent infrared radiation, having a total frequency range was from visible light down to [[extremely high frequency|millimeter waves]]. The Russian physicist [[Vitaly Ginzburg]] showed theoretically that undulators could be built in a 1947 paper. [[Julian Schwinger]] published a useful paper in 1949.<ref name="js">{{Cite doi|10.1103/PhysRev.75.1912}}</ref> that reduced the necessary calculations to [[Bessel functions]], for which there were tables. (The first [[UNIVAC I]] computer was not delivered until March 31, 1951.)

==References==
{{Reflist}}

==External links==
D. T. Attwood's page at Berkeley: [http://www.coe.berkeley.edu/AST/sxreuv/ Soft X-Rays and Extreme Ultraviolet Radiation]. His lecture and viewgraphs are available online.

[[Category:Synchrotron instrumentation]]

[[fr:Synchrotron#Éléments d'insertion]]

Ontology alignment

2013-10-01T23:54:52Z

68.65.169.68: /* Ontology alignment methods */

{{Merge into|Entropy|date=July 2013}}
{{Other uses|Statistical thermodynamics}}
In [[classical physics|classical]] [[statistical mechanics]], the [[entropy]] function earlier introduced by Clausius is changed to '''statistical entropy''' using [[probability theory]]. The statistical entropy perspective was introduced in 1870 with the work of the Austrian physicist [[Ludwig Boltzmann]].

== Gibbs Entropy Formula==
The macroscopic state of the system is defined by a distribution on the [[microstate (statistical mechanics)|microstates]] that are accessible to a system in the course of its [[thermal fluctuations]]. So the entropy is defined over two different levels of description of the given system. The entropy is given by the Gibbs entropy formula, named after [[Josiah Willard Gibbs|J. Willard Gibbs]]. For a classical system (i.e., a collection of classical particles) with a discrete set of microstates, if <math>E_i</math> is the energy of microstate ''i'', and <math>p_i</math> is its probability that it occurs during the system's fluctuations, then the entropy of the system is

: <math>S = -k_B\,\sum_i p_i \ln \,p_i</math>

<div style=" width: 320px; float: right; margin: 0 0 1em 1em; border-style: solid; border-width: 1px; padding: 1em; font-size: 90%">
'''Entropy changes for systems in a canonical state'''

A system with a well-defined temperature, i.e., one in thermal equilibrium with a thermal reservoir, has a probability of being in a microstate ''i'' given by [[Boltzmann's distribution]].

Changes in the entropy caused by changes in the external constraints are then given by:
:<math> dS = -k_B\,\sum_i dp_i \ln p_i</math>
:<math> \,\,\, = -k_B\,\sum_i dp_i (-E_i/k_BT -\ln Z)</math>
:<math> \,\,\, = \sum_i E_i dp_i / T </math>
:<math> \,\,\, = \sum_i [d (E_i p_i) - (dE_i) p_i] / T </math>

where we have twice used the conservation of probability, ''∑ dpi=0'' .

Now, ''∑i d (Ei pi)'' is the expectation value of the change in the total energy of the system.

If the changes are sufficiently slow, so that the system remains in the same microscopic state, but the state slowly (and reversibly) changes, then ''∑i (dEi) pi'' is the expectation value of the work done on the system through this reversible process, ''dwrev''.

But from the first law of thermodynamics, ''δE = δw +δq''. Therefore,
:<math>dS = \frac{\delta\langle q_{rev} \rangle}{T}</math>

In the [[thermodynamic limit]], the fluctuation of the macroscopic quantities from their average values becomes negligible; so this reproduces the definition of entropy from classical thermodynamics, given above.
</div>

The quantity <math>k_B</math> is a [[physical constant]] known as [[Boltzmann constant|Boltzmann's constant]], which, like the entropy, has units of [[heat capacity]]. The [[Natural logarithm|logarithm]] is [[Dimensionless number|dimensionless]].

This definition remains valid even when the system is far away from equilibrium. Other definitions assume that the system is in [[thermal equilibrium]], either as an [[isolated system]], or as a system in exchange with its surroundings. The set of microstates on which the sum is to be done is called a [[statistical ensemble]]. Each [[statistical ensemble]] (micro-canonical, canonical, grand-canonical, etc.) describes a different configuration of the system's exchanges with the outside, from an isolated system to a system that can exchange one more quantity with a reservoir, like energy, volume or molecules. In every ensemble, the [[thermodynamic equilibrium|equilibrium]] configuration of the system is dictated by the maximization of the entropy of the union of the system and its reservoir, according to the [[second law of thermodynamics]] (see the [[statistical mechanics]] article).

Neglecting [[correlation]]s between the different possible states (or, more generally, neglecting [[Statistical independence|statistical dependencies]] between states) will lead to an overestimate of the entropy.<ref name="jaynes1965">E.T. Jaynes; Gibbs vs Boltzmann Entropies; American Journal of Physics,391,1965</ref> These correlations occur in systems of interacting particles, that is, in all systems more complex than an [[ideal gas]].

This ''S'' is almost universally called simply the ''entropy''. It can also be called the ''statistical entropy'' or the ''thermodynamic entropy'' without changing the meaning. Note the above expression of the statistical entropy is a discretized version of [[Shannon entropy]]. The [[von Neumann entropy]] formula is an extension of the Gibbs entropy formula to the [[Quantum mechanics|quantum mechanical]] case.

It has been shown that the Gibb's Entropy is numerically equal to the experimental entropy<ref name="jaynes1965" /> <math>dS = \frac{\delta Q}{T} \!</math>

== Boltzmann's principle ==
{{main|Boltzmann's entropy formula}}
In Boltzmann's definition, entropy is a measure of the number of possible microscopic states (or '''microstates''') of a system in [[thermodynamic equilibrium]], consistent with its macroscopic thermodynamic properties (or '''macrostate'''). To understand what microstates and macrostates are, consider the example of a [[gas]] in a container. At a microscopic level, the gas consists of a [[Avogadro's number|vast number]] of freely moving [[atom]]s, which occasionally collide with one another and with the walls of the container. The microstate of the system is a description of the [[position (vector)|position]]s and [[momentum|momenta]] of all the atoms. In principle, all the physical properties of the system are determined by its microstate. However, because the number of atoms is so large, the motion of individual atoms is mostly irrelevant to the behavior of the system as a whole. Provided the system is in thermodynamic equilibrium, the system can be adequately described by a handful of macroscopic quantities, called "thermodynamic variables": the total [[energy]] ''E'', [[volume]] ''V'', [[pressure]] ''P'', [[temperature]] ''T'', and so forth. The macrostate of the system is a description of its thermodynamic variables.

There are three important points to note. Firstly, to specify any one microstate, we need to write down an impractically long list of numbers, whereas specifying a macrostate requires only a few numbers (''E'', ''V'', etc.). However, and this is the second point, the usual [[thermodynamic equations]] only describe the macrostate of a system adequately when this system is in equilibrium; non-equilibrium situations can generally ''not'' be described by a small number of variables. For example, if a gas is sloshing around in its container, even a macroscopic description would have to include, e.g., the velocity of the fluid at each different point. Actually, the macroscopic state of the system will be described by a small number of variables only if the system is at global [[thermodynamic equilibrium]]. Thirdly, more than one microstate can correspond to a single macrostate. In fact, for any given macrostate, there will be a huge number of microstates that are consistent with the given values of ''E'', ''V'', etc.

We are now ready to provide a definition of entropy. The entropy ''S'' is defined as

:<math>S = k_B \ln \Omega \!</math>
where
:''kB'' is [[Boltzmann constant|Boltzmann's constant]] and
:''<math>\Omega \!</math>'' is the number of microstates consistent with the given macrostate.

The statistical entropy reduces to Boltzmann's entropy when all the accessible microstates of the system are equally likely. It is also the configuration corresponding to the maximum of a system's entropy for a given set of accessible [[microstate (statistical mechanics)|microstates]], in other words the macroscopic configuration in which the lack of information is maximal. As such, according to the [[second law of thermodynamics]], it is the [[thermodynamic equilibrium|equilibrium]] configuration of an isolated system. Boltzmann's entropy is the expression of entropy at thermodynamic equilibrium in the micro-canonical ensemble.

This postulate, which is known as Boltzmann's principle, may be regarded as the foundation of [[statistical mechanics]], which describes thermodynamic systems using the statistical behaviour of its constituents. It turns out that ''S'' is itself a thermodynamic property, just like ''E'' or ''V''. Therefore, it acts as a link between the microscopic world and the macroscopic. One important property of ''S'' follows readily from the definition: since ''Ω'' is a [[natural number]] (1,2,3,...), ''S'' is either ''zero'' or ''positive'' (ln(1)=0, lnΩ≥0.)

=== Ensembles ===

The various ensembles used in statistical thermodynamics are linked to the entropy by the following relations:{{clarify|reason=What are the quantities that are being maintained constant between these different ensembles? Is this relationship only valid in the thermodynamic limit?|date=September 2013}}

:<math>S=k_B \ln \Omega_{\rm mic} = k_B (\ln Z_{\rm can} + \beta \bar E) = k_B (\ln \mathcal{Z}_{\rm gr} + \beta (\bar E - \mu \bar N)) </math>

<math>\Omega_{\rm mic} </math> is the [[microcanonical ensemble|microcanonical partition function]] 
<math>Z_{\rm can} </math> is the [[canonical ensemble|canonical partition function]] 
<math>\mathcal{Z}_{\rm gr} </math> is the [[grand canonical ensemble|grand canonical partition function]]

== Lack of knowledge and the second law of thermodynamics ==
We can view ''Ω'' as a measure of our lack of knowledge about a system. As an illustration of this idea, consider a set of 100 [[coin]]s, each of which is either [[coin flipping|heads up or tails up]]. The macrostates are specified by the total number of heads and tails, whereas the microstates are specified by the facings of each individual coin. For the macrostates of 100 heads or 100 tails, there is exactly one possible configuration, so our knowledge of the system is complete. At the opposite extreme, the macrostate which gives us the least knowledge about the system consists of 50 heads and 50 tails in any order, for which there are 100,891,344,545,564,193,334,812,497,256 ([[combination|100 choose 50]]) ≈ 1029 possible microstates.

Even when a system is entirely isolated from external influences, its microstate is constantly changing. For instance, the particles in a gas are constantly moving, and thus occupy a different position at each moment of time; their momenta are also constantly changing as they collide with each other or with the container walls. Suppose we prepare the system in an artificially highly ordered equilibrium state. For instance, imagine dividing a container with a partition and placing a gas on one side of the partition, with a vacuum on the other side. If we remove the partition and watch the subsequent behavior of the gas, we will find that its microstate evolves according to some chaotic and unpredictable pattern, and that on average these microstates will correspond to a more disordered macrostate than before. It is ''possible'', but ''extremely unlikely'', for the gas molecules to bounce off one another in such a way that they remain in one half of the container. It is overwhelmingly probable for the gas to spread out to fill the container evenly, which is the new equilibrium macrostate of the system.

This is an example illustrating the [[second law of thermodynamics|Second Law of Thermodynamics]]:

:''the total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value''.

Since its discovery, this idea has been the focus of a great deal of thought, some of it confused. A chief point of confusion is the fact that the Second Law applies only to ''isolated'' systems. For example, the [[Earth]] is not an isolated system because it is constantly receiving energy in the form of [[sunlight]]. In contrast, the [[universe]] may be considered an isolated system, so that its total disorder is constantly increasing.

== Counting of microstates ==
In [[classical mechanics|classical]] [[statistical mechanics]], the number of microstates is actually [[Uncountable set|uncountably infinite]], since the properties of classical systems are continuous. For example, a microstate of a classical ideal gas is specified by the positions and momenta of all the atoms, which range continuously over the [[real number]]s. If we want to define ''Ω'', we have to come up with a method of grouping the microstates together to obtain a countable set. This procedure is known as [[coarse graining]]. In the case of the ideal gas, we count two states of an atom as the "same" state if their positions and momenta are within ''δx'' and ''δp'' of each other. Since the values of ''δx'' and ''δp'' can be chosen arbitrarily, the entropy is not uniquely defined. It is defined only up to an additive constant. (As we will see, the [[Entropy (classical thermodynamics)|thermodynamic definition of entropy]] is also defined only up to a constant.)

This ambiguity can be resolved with [[quantum mechanics]]. The [[quantum state]] of a system can be expressed as a superposition of "basis" states, which can be chosen to be energy [[eigenstate]]s (i.e. eigenstates of the quantum [[Hamiltonian (quantum mechanics)|Hamiltonian]].) Usually, the quantum states are discrete, even though there may be an infinite number of them. For a system with some specified energy E, one takes Ω to be the number of energy eigenstates within a macroscopically small energy range between E and E + δE. In the [[thermodynamical limit]], the specific entropy becomes independent on the choice of δE.

An important result, known as [[Nernst's theorem]] or the [[third law of thermodynamics]], states that the entropy of a system at [[absolute zero|zero absolute temperature]] is a well-defined constant. This is because a system at zero temperature exists in its lowest-energy state, or [[ground state]], so that its entropy is determined by the [[Hamiltonian (quantum mechanics)|degeneracy]] of the ground state. Many systems, such as [[crystal|crystal lattices]], have a unique ground state, and (since ln(1) = 0) this means that they have zero entropy at absolute zero. Other systems have more than one state with the same, lowest energy, and have a non-vanishing "zero-point entropy". For instance, ordinary [[ice]] has a zero-point entropy of 3.41 J/(mol·K), because its underlying [[crystal structure]] possesses multiple configurations with the same energy (a phenomenon known as [[geometrical frustration]]).

The third law of thermodynamics states that the entropy of a perfect crystal at absolute zero, or 0 [[kelvin]] is zero. This means that in a perfect crystal, at 0 kelvin, nearly all molecular motion should cease in order to achieve ΔS=0. A perfect crystal is one in which the internal lattice structure is the same at all times; in other words, it is fixed and non-moving, and does not have rotational or vibrational energy. This means that there is only one way in which this order can be attained: when every particle of the structure is in its proper place.

However, the [[quantum harmonic oscillator|oscillator equation]] for predicting quantized vibrational levels shows that even when the vibrational quantum number is 0, the molecule still has vibrational energy. This means that no matter how cold the temperature gets, the lattice will always vibrate. This is in keeping with the Heisenberg uncertainty principle, which states that both the position and the momentum of a particle cannot be known precisely, at a given time:

:<math>E_\nu=h\nu_0(n+\begin{matrix} \frac{1}{2} \end{matrix}),</math>

where <math>h</math> is Planck's constant, <math>\nu_0</math> is the characteristic frequency of the vibration, and <math>n</math> is the vibrational quantum number. Note that even when <math>n=0</math> (the [[zero-point energy]]), <math>E_n</math> does not equal 0.

== See also ==
*[[Boltzmann constant]]
*[[Configuration entropy]]
*[[Conformational entropy]]
*[[Enthalpy]]
*[[Entropy]]
*[[Entropy (classical thermodynamics)]]
*[[Entropy (energy dispersal)]]
*[[Entropy of mixing]]
*[[Entropy (order and disorder)]]
*[[Entropy (information theory)]]
*[[History of entropy]]
*[[Information theory]]
*[[Thermodynamic free energy]]

== References ==
{{Reflist}}
*Boltzmann, Ludwig (1896, 1898). Vorlesungen über Gastheorie : 2 Volumes - Leipzig 1895/98 UB: O 5262-6. English version: Lectures on gas theory. Translated by Stephen G. Brush (1964) Berkeley: University of California Press; (1995) New York: Dover ISBN 0-486-68455-5

{{DEFAULTSORT:Entropy (Statistical Thermodynamics)}}
[[Category:Thermodynamic entropy]]