formulasearchengine - User contributions [en]

Resonant inductive coupling

2014-02-11T02:16:30Z

71.167.61.146:

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Color wheel graphs of complex functions

2014-01-19T18:18:15Z

71.167.61.80:

The '''random walker algorithm''' is an algorithm for [[image segmentation]]. In the first description of the algorithm,<ref name="grady2006random">L. Grady: [http://www.cns.bu.edu/~lgrady/grady2006random.pdf Random Walks for Image Segmentation], IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 28, No. 11, pp. 1768–1783, Nov., 2006.</ref> a user interactively labels a small number of pixels with known labels (called seeds), e.g., "object" and "background". The unlabeled pixels are each imagined to release a random walker, and the probability is computed that each pixel's random walker first arrives at a seed bearing each label, i.e., if a user places K seeds, each with a different label, then it is necessary to compute, for each pixel, the probability that a random walker leaving the pixel will first arrive at each seed. This computation may be determined analytically by solving a system of linear equations. After computing these probabilities for each pixel, the pixel is assigned to the label for which it is most likely to send a random walker. The image is modeled as a [[Graph (mathematics)|graph]], in which each pixel corresponds to a node which is connected to neighboring pixels by edges, and the edges are weighted to reflect the similarity between the pixels. Therefore, the random walk occurs on the weighted graph (see Doyle and Snell for an introduction to random walks on graphs<ref>P. Doyle, J. L. Snell: Random Walks and Electric Networks, Mathematical Association of America, 1984</ref>).

Although the initial algorithm was formulated as an interactive method for image segmentation, it has been extended to be a fully automatic algorithm, given a data fidelity term (e.g., an intensity prior).<ref name="grady2005multilabel">Leo Grady: Multilabel Random Walker Image Segmentation Using Prior Models, Proc. of CVPR, Vol. 1, pp. 763–770, 2005. [http://www.cns.bu.edu/%7Elgrady/grady2005multilabel.pdf]</ref> It has also been extended to other applications, such as Image Matching (R. Shen, I. Cheng, X.li and A. Basu), ICPR 2008, and Image Fusion, (R. Shen, I. Cheng, J.Shi and A. Basu), IEEE Trans. on Image Processing, 2011, and other applications.

The algorithm was initially published as a conference paper<ref>Leo Grady, Gareth Funka-Lea: Multi-Label Image Segmentation for Medical Applications Based on Graph-Theoretic Electrical Potentials, Proc. of the 8th ECCV Workshop on Computer Vision Approaches to Medical Image Analysis and Mathematical Methods in Biomedical Image Analysis, pp. 230–245, 2004.</ref> and later as a journal paper.<ref name="grady2006random" />

==Mathematics==

Although the algorithm was described in terms of random walks, the probability that each node sends a random walker to the seeds may be calculated analytically by solving a sparse, positive-definite system of linear equations with the graph [[Laplacian matrix of a graph|Laplacian matrix]], which we may represent with the variable <math>L</math>. The algorithm was shown to apply to an arbitrary number of labels (objects), but the exposition here is in terms of two labels (for simplicity of exposition).

Assume that the image is represented by a [[Graph (mathematics)|graph]], with each node <math>v_i</math> associated with a pixel and each edge <math>e_{ij}</math> connecting neighboring pixels <math>v_i</math> and <math>v_j</math>. The edge weights are used to encode node similarity, which may be derived from differences in image intensity, color, texture or any other meaningful features. For example, using image intensity <math>g_i</math> at node <math>v_i</math>, it is common to use the edge weighting function

:<math>w_{ij} = \exp{\left(-\beta (g_i - g_j)^2\right)}.
</math>

The nodes, edges and weights can then be used to construct the graph [[Laplacian matrix of a graph|Laplacian matrix]].

The random walker algorithm optimizes the energy
:<math>Q(x) = x^T L x = \sum_{e_{ij}} w_{ij} \left(x_i - x_j\right)^2</math>
where <math>x_i</math> represents a real-valued variable associated with each node in the graph and the optimization is constrained by <math>x_i = 1</math> for <math>v_i \in F</math> and <math>x_i = 0</math> for <math>v_i \in B</math>, where <math>F</math> and <math>B</math> represent the sets of foreground and background seeds, respectively. If we let <math>S</math> represent the set of nodes which are seeded (i.e., <math>S = F \cup B</math>) and <math>\overline{S}</math> represent the set of unseeded nodes (i.e., <math>S \cup \overline{S} = V</math> where <math>V</math> is the set of all nodes), then the optimum of the energy minimization problem is given by the solution to
:<math>
L_{\overline{S},\overline{S}} x_{\overline{S}} = - L_{\overline{S},S} x_{S},
</math>
where the subscripts are used to indicate the portion of the graph Laplacian matrix <math>L</math> indexed by the respective sets.

To incorporate likelihood (unary) terms into the algorithm, it was shown in <ref name="grady2005multilabel" /> that one may optimize the energy
:<math>Q(x) = x^T L x + \gamma \left((1-x)^T F (1-x) + x^T B x\right) = \sum_{e_{ij}} w_{ij} \left(x_i - x_j\right)^2 + \gamma \left(\sum_{v_i} f_i (1-x_i)^2 + \sum_{v_i} b_i x_i^2 \right),</math>
for positive, diagonal matrices <math>F</math> and <math>B</math>. Optimizing this energy leads to the system of linear equations
:<math>
\left(L_{\overline{S},\overline{S}} + \gamma F_{\overline{S},\overline{S}} + \gamma B_{\overline{S},\overline{S}}\right) x_{\overline{S}} = - L_{\overline{S},S} x_{S} - \gamma F_{\overline{S},\overline{S}}.
</math>
The set of seeded nodes, <math>S</math>, may be empty in this case (i.e., <math>\overline{S}=V</math>), but the presence of the positive diagonal matrices allows for a unique solution to this linear system.

For example, if the likelihood/unary terms are used to incorporate a color model of the object, then <math>f_i</math> would represent the confidence that the color at node <math>v_i</math> would belong to object (i.e., a larger value of <math>f_i</math> indicates greater confidence that <math>v_i</math> belonged to the object label) and <math>b_i</math> would represent the confidence that the color at node <math>v_i</math> belongs to the background.

==Algorithm interpretations==

The random walker algorithm was initially motivated by labeling a pixel as object/background based on the probability that a random walker dropped at the pixel would first reach an object (foreground) seed or a background seed. However, there are several other interpretations of this same algorithm which have appeared in.<ref name="grady2006random" />

===Circuit theory interpretations===

There are well-known connections between [[electrical circuit]] theory and random walks on graphs.<ref>P. G. Doyle, J. L. Snell: Random Walks and Electrical Networks, Carus Mathematical Monographs, 1984</ref> Consequently, the random walker algorithm has two different interpretations in terms of an electric circuit. In both cases, the graph is viewed as an electric circuit in which each edge is replaced by a passive linear [[resistor]]. The resistance, <math>r_{ij}</math>, associated with edge <math>e_{ij}</math> is set equal to <math>r_{ij} = \frac{1}{w_{ij}}</math> (i.e., the edge weight equals [[electrical conductance]]).

In the first interpretation, each node associated with a background seed, <math>v_i \in B</math>, is tied directly to [[Ground (electricity)|ground]] while each node associated with an object/foreground seed, <math>v_i \in F</math> is attached to a unit [[direct current]] ideal [[voltage source]] tied to ground (i.e., to establish a unit potential at each <math>v_i \in F</math>). The steady-state electrical circuit potentials established at each node by this circuit configuration will exactly equal the random walker probabilities. Specifically, the electrical potential, <math>x_i</math> at node <math>v_i</math> will equal the probability that a random walker dropped at node <math>v_i</math> will reach an object/foreground node before reaching a background node.

In the second interpretation, labeling a node as object or background by thresholding the random walker probability at 0.5 is equivalent to labeling a node as object or background based on the relative effective conductance between the node and the object or background seeds. Specifically, if a node has a higher effective conductance (lower effective resistance) to the object seeds than to the background seeds, then node is labeled as object. If a node has a higher effective conductance (lower effective resistance) to the background seeds than to the object seeds, then node is labeled as background.

==Extensions==

The traditional random walker algorithm described above has been extended in several ways:

* Random walks with restart<ref>T. H. Kim, K. M. Lee, S. U. Lee: Generative Image Segmentation Using Random Walks with Restart, Proc. of ECCV 2008, pp. 264–275</ref>
* Alpha matting<ref>J. Wang, M. Agrawala, M. F. Cohen: Soft scissors: an interactive tool for realtime high quality matting, Proc. of SIGGRAPH 2007</ref>
* Threshold selection<ref>S. Rysavy, A. Flores, R. Enciso, K. Okada: Classifiability Criteria for Refining of Random Walks Segmentation, Proc. of ICPR 2008</ref>
* Soft inputs<ref>W. Yang, J. Cai, J. Zheng, J. Luo: User-friendly Interactive Image Segmentation through Unified Combinatorial User Inputs, IEEE Trans. on Image Proc., 2010</ref>
* Run on a presegmented image<ref>C. Chefd'hotel, A. Sebbane: Random walk and front propagation on watershed adjacency graphs for multilabel image segmentation, Proc. of ICV 2007</ref>
* Scale space random walk<ref>R. Rzeszutek, T. El-Maraghi, D. Androutsos: Image segmentation using scale-space random walks, Proc. of the 16th international conference on Digital Signal Processing, pp. 458–461, 2009</ref>
* Fast random walker using offline [[precomputation]] <ref>L. Grady, A.K. Sinop: Fast approximate random walker segmentation using eigenvector
precomputation. In IEEE Conf. CVPR, pp. 1–8, 2008</ref><ref>S. Andrews, G. Hamarneh, A. Saad. Fast random walker with priors using precomputation for interactive medical image segmentation, Proc. of MICCAI 2010</ref>

==Applications==

Beyond image segmentation, the random walker algorithm has been additionally applied to several problems in computer vision and graphics:

* Image Colorization<ref>X. Liu, J. Liu, Z. Feng: Colorization Using Segmentation with Random Walk, Computer Analysis of Images and Patterns, pp. 468–475, 2009</ref>
* Interactive rotoscoping<ref>R. Rzeszutek, T. El-Maraghi, D. Androutsos: Interactive rotoscoping through scale-space random walks, Proc. of the 2009 IEEE international conference on Multimedia and Expo</ref>
* Medical image segmentation<ref>S. P. Dakua, J. S. Sahambi: LV Contour Extraction from Cardiac MR
Images Using Random Walks Approach, Int. Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009</ref><ref>F. Maier, A. Wimmer, G. Soza, J. N. Kaftan, D. Fritz, R. Dillmann: Automatic Liver Segmentation Using the Random Walker Algorithm, Bildverarbeitung für die Medizin 2008</ref><ref>P. Wighton, M. Sadeghi, T. K. Lee, M. S. Atkins: A Fully Automatic Random Walker Segmentation for Skin Lesions in a Supervised Setting, Proc. of MICCAI 2009</ref>
* Merging multiple segmentations<ref>P. Wattuya, K. Rothaus, J. S. Prassni, X. Jiang: A random walker based approach to combining multiple segmentations, Proc. of ICPR 2008</ref>
* Mesh segmentation<ref>Y.-K. Lai, S.-M. Hu, R. R. Martin, P. L. Rosin: Fast mesh segmentation using random walks, Proc. of the 2008 ACM symposium on Solid and physical modeling</ref><ref>J. Zhang, J. Zheng, J. Cai: Interactive Mesh Cutting Using Constrained Random Walks, IEEE Trans. on Visualization and Computer Graphics, 2010.</ref>
* Mesh denoising<ref>X. Sun, P. L. Rosin, R. R. Martin, F. C. Langbein: Random walks for feature-preserving mesh denoising, Computer Aided Geometric Design, Vol. 25, No. 7, Oct. 2008, pp. 437–456</ref>
* Segmentation editing<ref>L. Grady, G. Funka-Lea: An Energy Minimization Approach to the Data Driven Editing of Presegmented Images/Volumes, Proc. of MICCAI, Vol. 2, 2006, pp. 888–895</ref>
* Shadow elimination<ref>G. Li, L. Qingsheng, Q. Xiaoxu: Moving Vehicle Shadow Elimination Based on Random Walk and Edge Features, Proc. of IITA 2008</ref>
* Image matching<ref>R. Shen, I. Cheng, X. Li, A. Basu: Stereo matching using random walks, Proc. of ICPR 2008</ref>
* Image Fusion<ref>R. Shen, I. Cheng, J. Shi, A. Basu: Generalized Random Walks for Fusion of Multi-exposure Images, IEEE Trans. on Image Processing, 2011.</ref>

==References==

{{Reflist}}


==External links==
*[http://www.cns.bu.edu/~lgrady/random_walker_matlab_code.zip Matlab code implementing the original random walker algorithm]
*[http://fastrw.cs.sfu.ca/ Matlab code implementing the random walker algorithm with precomputation]
*[http://scikits-image.org/docs/dev/auto_examples/plot_random_walker_segmentation.html Python implementation of the original random walker algorithm] in the image processing toolbox [http://scikits-image.org/ scikits-image]

[[Category:Image segmentation]]

Electronic filter topology

2014-01-17T22:23:21Z

71.167.61.77:

'''Tarski–Grothendieck set theory''' ('''TG''', named after mathematicians [[Alfred Tarski]] and [[Alexander Grothendieck]]) is an [[axiomatic set theory]] that was introduced as part of the [[Mizar system]] for formal verification of proofs.

Tarski–Grothendieck set theory is a [[non-conservative extension]] of [[Zermelo–Fraenkel set theory]] (ZFC) and is distinguished from other axiomatic set theories by the inclusion of '''Tarski's axiom''' which states that for each set there is a [[Grothendieck universe]] it belongs to (see below). Tarski's axiom implies the existence of [[inaccessible cardinal]]s, providing a richer [[ontology]] than that of conventional set theories such as ZFC.

==Axioms==

While the [[axiom]]s and [[definition]]s defining Mizar's basic objects and processes are fully [[Formal system|formal]], they are described informally below.

* Given any set <math>A</math>, the singleton <math>\{A\}</math> exists.
* Given any two sets, their unordered and ordered pairs exist.
* Given any family of sets, its union exists.

'''TG''' includes the following axioms, which are conventional because also part of [[ZFC]]:
* Set axiom: Quantified variables range over sets alone; everything is a set (the same [[ontology]] as [[ZFC]]).
* [[Extensionality]] axiom: Two sets are identical if they have the same members.
* [[Axiom of regularity]]: No set is a member of itself, and circular chains of membership are impossible.
* [[Axiom schema of replacement]]: Let the [[domain (mathematics)|domain]] of the [[function (mathematics)|function]] <math>F</math> be the set <math>A</math>. Then the [[range (mathematics)|range]] of <math>F</math> (the values of <math>F(x)</math> for all members <math>x</math> of <math>A</math>) is also a set.

It is Tarski's axiom that distinguishes '''TG''' from other axiomatic set theories. Tarski's axiom also implies the axioms of [[axiom of infinity|infinity]], [[axiom of choice|choice]],<ref>Tarski (1938)</ref><ref>http://mmlquery.mizar.org/mml/current/wellord2.html#T26</ref> and [[axiom of power set|power set]].<ref>Robert Solovay, [http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html Re: AC and strongly inaccessible cardinals].</ref><ref>[http://us.metamath.org/mpegif/grothpw.html Metamath '''grothpw'''.]</ref> It also implies the existence of [[inaccessible cardinal]]s, thanks to which the [[ontology]] of '''TG''' is much richer than that of conventional set theories such as [[ZFC]].

* '''Tarski's axiom''' (adapted from Tarski 1939<ref>Tarski (1939)</ref>). For every set <math>x</math>, there exists a set <math>y</math> whose members include:

- <math>x</math> itself;

- every subset of every member of <math>y</math>;

- the power set of every member of <math>y</math>;

- every subset of <math>y</math> of [[cardinality]] less than that of <math>y</math>.

More formally:

:<math>\exists y [x\in y \wedge \forall z\in y(\mathcal P(z)\subseteq y\wedge\mathcal P(z)\in y) \wedge \forall z\in\mathcal P(y)(\neg z\approx y\to z\in y)]</math>

where "<math>\mathcal P(x)</math>" denotes the power class of ''x'' and "<math>\approx</math>" denotes [[equinumerosity]]. What Tarski's axiom states (in the vernacular) for each set <math>x</math> there is a [[Grothendieck universe]] it belongs to.

==Implementation in the Mizar system==

The Mizar language, underlying the implementation of '''TG''' and providing its logical syntax, is typed and the types are assumed to be non-empty. Hence, the theory is implicitly taken to be [[Axiom of empty set|non-empty]]. The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors.

The system includes equality, the membership predicate and the following standard definitions:
* [[Singleton (mathematics)|Singleton]]: A set with one member;
* [[Unordered pair]]: A set with two distinct members. <math>\{a,b\} = \{b,a\}</math>;
* [[Ordered pair]]: The set <math>\{\{a,b\},\{a\}\} = (a,b) \neq (b,a)</math>;
* [[Subset]]: A set all of whose members are members of another given set;
* The [[union (set theory)|union]] of a family of sets <math>Y</math>: The set of all members of every member of <math>Y</math>.

==See also==
*[[Mizar system]]
*[[Grothendieck universe]]
*[[Axiom of limitation of size]]

==Notes==
<references/>

==References==
*Andreas Blass, I.M. Dimitriou, and [[Benedikt Löwe]] (2007) "[http://dare.uva.nl/document/25381 Inaccessible Cardinals without the Axiom of Choice,]" ''Fundamenta Mathematicae'' 194: 179-89.
* {{cite conference
| first = Nicolas
| last = Bourbaki
| authorlink = Nicolas Bourbaki
| year = 1972
| title = Univers
| booktitle = Séminaire de Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 1 (Lecture notes in mathematics '''269''')
| editor = [[Michael Artin]], [[Alexandre Grothendieck]], [[Jean-Louis Verdier]], eds.
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| location = Berlin; New York
| language = French
| pages = 185–217
| url = http://modular.fas.harvard.edu/sga/sga/4-1/4-1t_185.html
}}
* [[Patrick Suppes]] (1960) ''Axiomatic Set Theory''. Van Nostrand. Dover reprint, 1972.
* {{cite journal
| last = Tarski
| first = Alfred
| authorlink = Alfred Tarski
| year = 1938
| title = Über unerreichbare Kardinalzahlen
| journal = Fundamenta Mathematicae
| volume = 30
| pages = 68–89
| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm30/fm30113.pdf
}}
* {{cite journal
| last = Tarski
| first = Alfred
| authorlink = Alfred Tarski
| year = 1939
| title = On the well-ordered subsets of any set
| journal = Fundamenta Mathematicae
| volume = 32
| pages = 176–183
| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm32/fm32115.pdf
}}

==External links==
*Trybulec, Andrzej, 1989, "[http://mizar.uwb.edu.pl/JFM/Axiomatics/tarski.html Tarski–Grothendieck Set Theory]", ''Journal of Formalized Mathematics''.
*[[Metamath]]: "[http://us.metamath.org/mpegif/mmset.html Proof Explorer Home Page.]" Scroll down to "Grothendieck's Axiom."
* [[PlanetMath]]: "[http://planetmath.org/encyclopedia/TarskisAxiom.html Tarski's Axiom]"

{{DEFAULTSORT:Tarski-Grothendieck set theory}}
[[Category:Systems of set theory]]

Sallen–Key topology

2014-01-17T15:46:32Z

71.167.61.77: roman labels, subscript inside superscript

[[Image:Hot metalwork.jpg|250px|thumb|right|Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy would ideally be the amount of heat required to warm the metal to its temperature, but this quantity is not well-defined, as there are many ways to obtain a given body at a given temperature, and each of them may require a different amount of total [[heat]] input. Thermal energy, unlike [[internal energy]], is therefore not a state function.]]
'''Thermal energy''' is the part of the total potential [[energy]] and kinetic energy of an [[Physical body|object]] or sample of matter that results in the system [[temperature]].<ref name=eb>[http://www.britannica.com/eb/article-9072068/thermal-energy Thermal energy entry in Britannica Online]</ref> It is represented by the variable Q, and can be measured in [[Joule]]s. This quantity may be difficult to determine or even meaningless unless the system has attained its temperature only through warming (heating), and not been subjected to work input or output, or any other energy-changing processes. Because the total amount of [[heat]] that enters an object is not a conserved quantity like mass or energy, and may be destroyed or created by many processes, the idea of an object's thermal energy or "heat content," something that remains a measureable and objective part of the [[internal energy]] of a body, cannot be strictly upheld. The idea of a thermal (part) of object internal energy is therefore useful only as an ideal model, in special cases where the total integrated energy of heat added or removed from a system happens to stay approximately constant as heat is conducted through the system.

The [[internal energy]] of a system, also often called the thermodynamic energy, includes other forms of energy in a thermodynamic system in addition to thermal energy, namely forms of [[potential energy]] that do not influence temperature and do not absorb heat, such as the [[chemical energy]] stored in its molecular structure and electronic configuration, and the nuclear [[binding energy]] that binds the sub-atomic particles of matter.

Microscopically, the thermal energy may include both the [[kinetic energy]] and [[potential energy]] of a system's constituent particles, which may be atoms, molecules, electrons, or particles in plasmas. It originates from the individually random, or disordered, motion of particles in a large ensemble, as consequence of absorbing [[heat]]. In ideal monatomic gases, thermal energy is entirely kinetic energy. In other substances, in cases where some of thermal energy is stored in atomic vibration, this vibrational part of the thermal energy is stored equally partitioned between potential energy of atomic vibration, and kinetic energy of atomic vibration. Thermal energy is thus [[equipartition theorem|equally partitioned]] between all available quadratic [[degrees of freedom (physics and chemistry)|degrees of freedom]] of the particles. As noted, these degrees of freedom may include pure translational motion in gases, in rotational states, and as potential and kinetic energy in [[normal mode]]s of vibrations in intermolecular or crystal [[lattice vibration]]s. In general, due to quantum mechanical reasons, the availability of any such degrees of freedom is a function of the energy in the system, and therefore depends on the temperature (see [[heat capacity]] for discussion of this phenomenon).

Macroscopically, the thermal energy of a system at a given temperature is related proportionally to its [[heat capacity]]. However, since the heat capacity differs according to whether or not constant volume or constant pressure is specified, or phase changes permitted, the heat capacity cannot be used define thermal energy unless it is done in such a way as to insure that only heat gain or loss (not work) makes any changes in the internal energy of the system. Usually, this means specifying the "constant volume heat capacity" of the system so that no work is done. Also the heat capacity of a system for such purposes must not include heat absorbed by any chemical reaction or process.

As noted, thermal energy is not a state function, or a property of a system, since the total thermal energy needed to warm a system to a given temperature depends on the path taken to attain the temperature, unless all forms of work and chemical potential change in the system are zero or negligible (in which case thermal energy is a subset of the internal energy). Thus, thermal energy is process-dependent except in systems in which processes to change internal energy other than heating, can be neglected. Nevertheless, when this is true, thermal energy and heat capacity may be a useful concept in the study of heat transfer in solids and liquids, in engineering and other disciplines.

== Differentiation from heat ==
[[Heat]], in the strict use in physics, is characteristic only of a process, i.e. it is absorbed or produced as an energy ''exchange'', always as a result of a temperature difference. Heat is thermal energy in the process of transfer or conversion across a boundary of one region of matter to another, as a result of a temperature difference.<ref name=speyer>{{cite book
|title=Thermal Analysis of Materials
|author=Robert F. Speyer
|publisher=Marcel Dekker, Inc.
|year=2012
|isbn=0-8247-8963-6
|series=Materials Engineering
|page=2
}}</ref> In engineering, the terms "heat" and "heat transfer" are thus used nearly interchangeably, since heat is always understood to be in the process of transfer. The energy transferred by heat is called by other terms (such as thermal energy or latent energy) when this energy is no longer in net transfer, and has become static.<ref name="Incrop">{{cite book
| author = Frank P. Incropera
| authorlink = Frank P. Incropera
| coauthors = David P. De Witt and D. P. Dewitt
| title = Fundamentals of Heat and Mass Transfer
| edition = 3rd
| publisher = [[John Wiley & Sons]]
| year = 1990
| page = 2
| isbn = 0-471-51729-1}} See box definition: "Heat transfer (or heat) is energy in transit due to a temperature difference." See page 14 for the definition of the thermal component of the thermodynamic [[internal energy]].</ref> Thus, heat is not a static property of matter. Matter does not contain heat, but rather thermal energy, and even the thermal energy is subject to transformations into and out of other types of energy, and so can be considered to be "conserved" only when these processes are small. The heat transfer rate or heating rate is the amount of energy per unit time being transferred as heat, or the heat [[power (physics)|power]].

When two thermodynamic systems with different temperatures are brought into diathermic contact, they spontaneously exchange energy as heat, the exchange being transfer of thermal energy from the system of higher temperature to the colder system. Heat may cause work to be performed on a system, for example, in form of volume or pressure changes. This work may be used in heat engines to convert thermal energy into other forms of energy. When two systems have reached a [[thermodynamic equilibrium]], they have attained the same exact temperature and the net exchange of thermal energy vanishes, and heat flow ceases.

==Definitions==
Thermal energy is the portion of the thermodynamic or internal energy of a system that is responsible for the temperature of the system.<ref name=eb /><ref name=speyer /> The thermal energy of a system scales with its size and is therefore an [[extensive property]]. It is not a [[state function]] of the system unless the system has been constructed so that all changes in internal energy are due to changes in thermal energy, as a result of heat transfer (not work). Otherwise thermal energy is dependent on the way or method by which the system attained its temperature.

From a macroscopic thermodynamic description, the thermal energy of a system is given by its constant volume specific [[heat capacity]] ''C(T)'', a temperature coefficient also called thermal capacity, at any given [[absolute temperature]] (''T''):
:<math>U_{thermal} = C(T) \cdot T.</math>
The heat capacity is a function of temperature itself, and is typically measured and specified for certain standard conditions and a specific [[amount of substance]] (molar heat capacity) or [[mass]] units (specific heat capacity). At constant volume (''V''), ''C''V it is the temperature coefficient of energy.<ref name=fuchs>{{cite book
|title=The Dynamics of Heat: A Unified Approach to Thermodynamics and Heat Transfer
|author=Hans U. Fuchs
|edition=2
|publisher=Springer
|year=2010
|isbn=978-1-4419-7603-1
|page=211
}}</ref> In practice, given a narrow temperature range, for example the operational range of a heat engine, the heat capacity of a system is often constant, and thus thermal energy changes are conveniently measured as temperature fluctuations in the system.

In the microscopical description of [[statistical physics]], the thermal energy is identified with the mechanical kinetic energy of the constituent particles or other forms of kinetic energy associated with quantum-mechanical [[Microstate (statistical mechanics)|microstates]].

The distinguishing difference between the terms ''kinetic energy'' and ''thermal energy'' is that thermal energy is the ''mean'' energy of disordered, i.e. random, motion of the particles or the oscillations in the system. The conversion of energy of ordered motion to thermal energy results from collisions.<ref name=blundell>{{cite book
|author=S. Blundell, K. Blundell
|title=Concepts in Thermal Physics
|year=2006
|publisher=Oxford University Press
|isbn=0-19-856769-3
|page=366
}}</ref>

All kinetic energy is partitioned into the degrees of freedom of the system. The average energy of a single particle with ''f'' quadratic degrees of freedom in a thermal bath of temperature ''T'' is a statistical mean energy given by the [[equipartition theorem]] as
:<math>E_{thermal} = f \cdot \tfrac 1 2 kT \,\!</math>
where ''k'' is the [[Boltzmann constant]]. The total thermal energy of a sample of matter or a thermodynamic system is consequently the average sum of the kinetic energies of all particles in the system. Thus, for a system of ''N'' particles its thermal energy is<ref name=schroeder>{{cite book
|title=An Introduction to Thermal Physics
|author=D.V. Schroeder
|publisher=Addison-Wesley
|year=1999
|isbn=0-201-38027-7
|page=15
}}</ref>
:<math>U_{thermal} = N \cdot f \cdot \tfrac{1}{2} kT.</math>
For gaseous systems, the factor ''f'', the number of degrees of freedom, commonly has the value 3 in the case of the monatomic gas, 5 for many diatomic gases, and 7 for larger molecules at ambient temperatures. In general however, it is a function of the temperature of the system as internal modes of motion, vibration, or rotation become available in higher energy regimes.

''Uthermal'' is not the total energy of a system. Physical systems also contains static [[potential energy]] (such as [[chemical energy]]) that arises from interactions between particles, [[Nuclear potential energy|nuclear energy]] associated with atomic nuclei of particles, and even the [[rest mass energy]] due to the equivalence of energy and mass.

==Thermal energy of the ideal gas==
Thermal energy is most easily defined in the context of the [[ideal gas]], which is well approximated by a [[monatomic]] gas at low pressure. The ideal gas is a gas of particles considered as point objects of perfect spherical symmetry that interact only by elastic collisions and fill a volume such that their mean free path between collisions is much larger than their diameter.

The mechanical kinetic energy of a single particle is
:<math>E_{kinetic} = \tfrac 1 2 m v^2 \,\!</math>
where ''m'' is the particle's mass and ''v'' is its velocity. The thermal energy of the gas sample consisting of ''N'' atoms is given by the sum of these energies, assuming no losses to the container or the environment:
:<math>U_{thermal} = \tfrac 1 2 N m \overline{v^2} = \tfrac{3}{2} N k T,</math>
where the line over the velocity term indicates that the average value is calculated over the entire ensemble. The total thermal energy of the sample is proportional to the macroscopic temperature by a constant factor accounting for the three translational degrees of freedom of each particle and the Boltzmann constant. The Boltzmann constant converts units between the microscopic model and the macroscopic temperature. This formalism is the basic assumption that directly yields the [[ideal gas law]] and it shows that for the ideal gas, the internal energy ''U'' consists only of its thermal energy:
:<math> U = U_{thermal}.\;</math>

==Historical context==
In an 1847 lecture entitled ''On Matter, Living Force, and Heat'', [[James Prescott Joule]] characterized various terms that are closely related to thermal energy and heat.
He identified the terms [[latent heat]] and [[sensible heat]] as forms of heat each effecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively.<ref>{{citation
|author=J. P. Joule
|title=Matter, Living Force, and Heat
|work=The Scientific Papers of James Prescott Joule
|year=1884
|publisher=The Physical Society of London
|page=274
|quote=I am inclined to believe that both of these hypotheses will be found to hold good,—that in some instances, particularly in the case of ''sensible'' heat, or such as is indicated by the thermometer, heat will be found to consist in the living force of the particles of the bodies in which it is induced; whilst in others, particularly in the case of ''latent'' heat, the phenomena are produced by the separation of particle from particle, so as to cause them to attract one another through a greater space.
|url=http://www.archive.org/details/scientificpapers01joul
|accessdate=2 January 2013}}
</ref>
He describes latent energy as the energy of interaction in a given configuration of particles, i.e. a form of [[potential energy]], and the sensible heat as an energy affecting temperature measured by the thermometer due to the thermal energy, which he called the ''living force''.

==Distinction of thermal energy and heat==

In thermodynamics, [[heat]] must always be defined as energy in exchange between two systems, or a single system and its surroundings.<ref name=leland>{{citation
|title=Basic Principles of Classical and Statistical Thermodynamics
|author=Thomas W. Leland, Jr.
|editor=G. A. Mansoori
|url=http://www.uic.edu/labs/trl/1.OnlineMaterials/BasicPrinciplesByTWLeland.pdf
}}</ref> According to the [[zeroth law of thermodynamics]], heat is exchanged ''between'' thermodynamic systems in thermal contact only if their temperatures are different, as this is the condition when the net exchange of thermal energy is non-zero. For the purpose of distinction, a system is defined to be enclosed by a well-characterized boundary. If heat traverses the boundary in direction ''into'' the system, the internal energy change is considered to be a positive quantity, while ''exiting'' the system, it is negative. As a process variable, heat is never a property of the system, nor is it ''contained'' within the boundary of the system.<ref name=speyer/>

In contrast to heat, thermal energy exists on both sides of a boundary. It is the statistical mean of the microscopic fluctuations of the kinetic energy of the systems' particles, and it is the source and the effect of the transfer of heat across a system boundary. Statistically, thermal energy is always exchanged between systems, even when the temperatures on both sides is the same, i.e. the systems are in thermal equilibrium. However, at equilibrium, the ''net'' exchange of thermal energy is zero, and therefore there is no heat.

Thermal energy may be increased in a system by other means than heat, for example when mechanical or electrical work is performed on the system. No qualitative difference exists between the thermal energy added by other means. Thermal energy is not a state function, although it may be closely related to the [[internal energy]] of some systems, which is a state function. There is also no need in classical thermodynamics to characterize the thermal energy in terms of atomic or molecular behavior. A change in thermal energy induced in a system is the product of the change in entropy and the temperature of the system.

Heat exchanged across a boundary may cause changes other than a change in temperature. For example, it may cause phase transitions, such as melting or evaporation, which are changes in the configuration of a material. Since such an energy exchange is not observable by a change in temperature, it is called a [[latent heat]] and represents a change in the potential energy of the system.

Rather than being itself the thermal energy involved in a transfer, heat is sometimes also understood as the process of that transfer, i.e. ''heat'' functions as a verb.

Today's narrow definition of ''heat'' in physics contrasts with its use in common language, in some engineering disciplines, and in the historical scientific development of thermodynamics in the [[caloric theory]] of heat. The phenomenon of ''heat'' in these instances is today properly identified as the [[entropy]].<ref name=fuchs/>

== The origin of heat energy on Earth ==



[[Earth|Earth's]] proximity to the [[Sun]] is the reason that almost everything near Earth's surface is warm with a temperature substantially above absolute zero.<ref>The deepest ocean depths (3 to 10 km) are no colder than about 274.7–275.7 K (1.5–2.5 °C). Even the world-record cold surface temperature established on July 21, 1983 at [[Vostok Station]], Antarctica is 184 K (a reported value of −89.2 °C). The residual heat of gravitational contraction left over from earth's formation, tidal friction, and the decay of radioisotopes in earth's core provide insufficient heat to maintain earth's surface, oceans, and atmosphere "substantially above" absolute zero in this context. Also, the qualification of "most-everything" provides for the exclusion of lava flows, which derive their temperature from these deep-earth sources of heat.</ref> [[Solar radiation]] constantly replenishes heat energy that Earth loses into space and a relatively stable state of near equilibrium is achieved. Because of the wide variety of heat diffusion mechanisms (one of which is black-body radiation which occurs at the speed of light), objects on Earth rarely vary too far from the global mean surface and air temperature of 287 to 288 K (14 to 15 °C). The more an object's or system's temperature varies from this average, the more rapidly it tends to come back into equilibrium with the ambient environment.

==Thermal energy of individual particles==
The term ''thermal energy'' is also often used as a property of single particles to designate the kinetic energy of the particles. An example is the description of [[thermal neutron]]s having a certain thermal energy, which means that the kinetic energy of the particle is equivalent to the temperature of its surroundings.

==See also==
*[[Heat transfer]]
*[[Ocean thermal energy conversion]]
*[[Thermal science]]

==References==
{{reflist}}

==External links==
*[http://solar.calfinder.com/library/thermal Example of incorrect use of ''heat'' and ''thermal energy'']

[[Category:Thermodynamics]]
[[Category:Forms of energy]]