Wikipedia - Benutzerbeiträge [de]

Entropische Gravitation

2010-09-24T03:52:48Z

AugPi: typo fix

[[Image:NewtonsLawOfUniversalGravitation.svg|thumb|right|200px|Verlinde's statistical description of gravity as an entropic force leads to the correct [[Newton's law of universal gravitation|inverse square distance law of attraction between classical bodies]].]]

The hypothesis of '''gravity being an entropic force''', also called '''entropic gravity''', has a history that goes back to research on [[black hole]] thermodynamics by [[Jacob Bekenstein|Bekenstein]] and [[Stephen Hawking|Hawking]] in the mid-1970s. These studies suggest a deep connection between [[gravity]] and thermodynamics, which describes the behavior of heat and gases. In 1995, [[Theodore Jacobson|Jacobson]] demonstrated that the [[Einstein equations]] describing relativistic gravitation can be derived by combining general thermodynamic considerations with the [[equivalence principle]].<ref>{{cite web|last=Jacobson|first=Theodore|title=Thermodynamics of Spacetime: The Einstein Equation of State|url=http://arxiv.org/abs/gr-qc/9504004|publisher=arXiv.org|accessdate=6 September 2010|doi=10.1103/PhysRevLett.75.1260|date=4 April 1995}}</ref> Subsequently, other physicists began to explore links between gravity and [[entropy]].<ref>{{cite web|last=Padmanabhan|first=Thanu|title=Thermodynamical Aspects of Gravity: New insights|url=http://arxiv.org/abs/0911.5004|publisher=arXiv.org|accessdate=6 September 2010|date=26 November 2009}}</ref>

In 2009, [[Erik Verlinde]] disclosed a conceptual theory that describes gravity as an entropic force.<ref>{{cite news|last=van Calmthout|first=Martijn|title=Is Einstein een beetje achterhaald?|url=http://www.volkskrant.nl/wetenschap/article1326775.ece/Is_Einstein_een_beetje_achterhaald|accessdate=6 September 2010|newspaper=de Volkskrant|date=12 December 2009|language=Dutch}}</ref> On January 6, 2010 he published a preprint of a 29 page paper titled "On the Origin of Gravity and the Laws of Newton".<ref name="VerlindePaper">{{cite web|last=Verlinde|first=Eric|title=Title: On the Origin of Gravity and the Laws of Newton|url=http://arxiv.org/abs/1001.0785|publisher=arXiv.org|accessdate=6 September 2010|date=6 January, 2010}}</ref> Reversing the logic of over 300 years, it argued that gravity is a consequence of the laws of thermodynamics. This theory combines the thermodynamic approach to gravity with [[Gerardus 't Hooft]]'s [[holographic principle]]. If proven correct, this implies gravity is not a [[fundamental interaction]], but an [[emergent phenomenon]] which arises from the statistical behavior of microscopic [[Degrees of freedom (physics and chemistry)|degrees of freedom]] encoded on a holographic screen. The paper drew a variety of responses from the scientific community. [[Andrew Strominger]], a string theorist at Harvard said “Some people have said it can’t be right, others that it’s right and we already knew it — that it’s right and profound, right and trivial."<ref name="nytimes">{{cite news|last=Overbye|first=Dennis|title=A Scientist Takes On Gravity|url=http://www.nytimes.com/2010/07/13/science/13gravity.html?_r=1|accessdate=6 September 2010|newspaper=[[The New York Times]]|date=July 12 2010}}</ref>

Verlinde's article also attracted a large amount of media exposure,<ref>[http://www.newscientist.com/article/mg20527443.800-the-entropy-force-a-new-direction-for-gravity.html?page=1 The entropy force: a new direction for gravity], [[New Scientist]], 20 January 2010, issue 2744</ref><ref>[http://www.wired.com/beyond_the_beyond/2010/01/gravity-is-an-entropic-form-of-holographic-information/ Gravity is an entropic form of holographic information], ''[[Wired Magazine]]'', 20 January 2010</ref> and led to immediate follow-up work in cosmology,<ref>[http://arxiv.org/abs/1001.3237v1 Equipartition of energy and the first law of thermodynamics at the apparent horizon], Fu-Wen Shu, Yungui Gong, 2010</ref><ref>[http://arxiv.org/abs/1001.3470v1 Friedmann equations from entropic force], Rong-Gen Cai, Li-Ming Cao, Nobuyoshi Ohta 2010</ref> the [[dark energy|dark energy hypothesis]],<ref>[http://www.scientificblogging.com/hammock_physicist/it_bit_how_get_rid_dark_energy It from Bit: How to get rid of dark energy], Johannes Koelman, 2010</ref> [[Metric expansion of space|cosmological acceleration]],<ref>[http://arXiv.org/abs/1002.4278 Entropic Accelerating Universe], Damien Easson, Paul Frampton, George Smoot, 2010</ref><ref>[http://arxiv.org/abs/1003.4526 Entropic cosmology: a unified model of inflation and late-time acceleration], Yi-Fu Cai, Jie Liu, Hong Li, 2010</ref> [[Inflation (cosmology)|cosmological inflation]],<ref>[http://arxiv.org/abs/1001.4786v1 Towards a holographic description of inflation and generation of fluctuations from thermodynamics], Yi Wang, 2010</ref> and [[loop quantum gravity]].<ref>[http://arxiv.org/abs/1001.3668v1 Newtonian gravity in loop quantum gravity], [[Lee Smolin]], 2010</ref> Also, a specific microscopic model has been proposed that indeed leads to entropic gravity emerging at large scales.<ref>[http://arxiv.org/abs/1001.3808v1 Notes concerning "On the origin of gravity and the laws of Newton" by E. Verlinde], Jarmo Makela, 2010</ref>

==Informal summary==
{{expert|Physics}}
Basically this theory says gravity is not an original force, but a resulting effect. This is based on the theoretical idea that the surface of a black hole contains all the information from the 3 dimensional world falling onto it. Its surface is 2 dimensional, there is no inside only the outer surface can contain (record) the information from everything that falls on it, in a sense if things fall on this surface they are converted to 2 dimensions. Now take that simple idea, as in physics formulas work in both directions of the time lets do this in reverse. Play a time movie of a black hole in reverse. In reverse time this 2 dimensional recording sends back gravity and the 3rd dimension. So if that 2 dimensional state is a perfect holographic recording of everything falling onto it, then gravity is the distortion of that perfect representation. Gravity kicks it out of such a perfect state, so its a entropic effect resulting in the 3rd dimension.

==See also==
*[[Entropic Spacetime Theory]]
*[[Ideal_chain#Entropic_elasticity_of_an_ideal_chain|Entropic elasticity of an ideal chain]]
*[[Entropic force]]
*[[Gravitation]]
*[[Induced gravity]]

==References==
{{Reflist}}

==Further reading==
*[http://www.science20.com/hammock_physicist/it_bit_entropic_gravity_pedestrians It from bit - Entropic gravity for pedestrians], J. Koelman
*[http://www.imsc.res.in/~iagrg/IagrgSite/Activities/IagrgMeetings/25th_Iagrg/VRtalk.pdf Gravity: the inside story], T Padmanabhan

{{Use dmy dates|date=August 2010}}
{{Theories of gravitation}}

{{DEFAULTSORT:Gravity As An Entropic Force}}
[[Category:Theories of gravitation]]
[[Category:Information theory]]
[[Category:Thermodynamics]]

[[zh:引力的熵力假说]]

Cayleygraph

2006-06-06T19:57:42Z

AugPi: /* Another example */ group presentation

[[Image:Cayley graph of F2.svg|right|thumb|The Cayley graph of the [[free group]] on two generators ''a'' and ''b'']]

In [[mathematics]], a '''Cayley graph''', named after [[Arthur Cayley]], is a [[graph theory|graph]] that encodes the structure of a [[group (mathematics)|group]]. It is a central tool in [[combinatorial group theory|combinatorial]] and [[geometric group theory]].

Let <math>G</math> be a group, and let <math>S</math> be a [[generating set of a group|generating set]] for <math>G</math>. The Cayley graph of <math>G</math> with respect to <math>S</math> has a vertex for every element of <math>G</math>, with an edge from <math>g</math> to <math>gs</math> for all elements <math>g\in G</math> and <math>s\in S</math>.

== Example ==
For example, the Cayley graph of the [[free group]] on two generators <math>a</math> and <math>b</math> is depicted above and to the right. (Note that <math>e</math> represents the [[identity element]].) Travelling right along an edge represents multiplying on the right by <math>a</math>, while travelling up corresponds to multiplying by <math>b</math>. Since the free group has no [[relation (mathematics)|relation]]s, the graph has no cycles.

== Another example ==
[[Image:CayleyGraphOfDihedralGroupD4.PNG|300px|right|thumb|The Cayley graph of the dihedral group D4 on two generators α and β]]
The Cayley graph of the [[dihedral group]] ''D''4 on two generators α and β is depicted to the right. Red arrows represent left-multiplication by element α. Since element β is [[Cayley table|self-inverse]], the blue lines which represent left-multiplication by element β are undirected. Therefore the graph is mixed: it has eight vertices, eight arrows, and four edges.

The [[Cayley table]] of the group ''D''4 can be derived from the [[presentation of a group|group presentation]]
<math> \langle \alpha, \beta | \alpha^4 = \beta^2 = e, \alpha \beta = \beta \alpha^3 \rangle </math>.

== Variations ==
The above definition gives a connected, directed graph. There are a number of slight variations on the definition:

#Usually <math>S</math> is not allowed to contain the identity element <math>e</math>.
#If the set <math>S</math> doesn't generate the whole group, the Cayley graph isn't connected.
#In some contexts, left multiplication is used instead of right. That is, edges go from <math>g</math> to <math>sg</math>.
#In many contexts, the generating set is assumed to be ''symmetric'', meaning that <math>s^{-1}</math> is in <math>S</math> whenever <math>s</math> is. In this case, the graph is undirected.

== The Sabidussi Theorem ==
<math>G</math> [[group action|acts]] on itself by multiplication on the left. This action induces an action of <math>G</math> on its Cayley graph. Explicitly, an element <math>h</math> sends a vertex <math>g</math> to the vertex <math>hg</math>, and the edge <math>(g,gs)</math> to the edge <math>(hg,hgs)</math>. Since the action of <math>G</math> on itself is [[transitive]], any Cayley graph is [[vertex-transitive]]. The [[Sabidussi theorem]] gives a characterization of Cayley graphs:
Graph <math>X</math> is a Cayley graph if and only if the automorphism group of <math>X</math> contains a subgroup <math>G</math> acting regularly on the vertex set of <math>X</math>.

== Schreier coset graph ==
If one, instead, takes the vertices to be right cosets of a fixed subgroup <math>H</math>, one obtains a related construction, the [[Schreier coset graph]], which is at the basis of [[coset enumeration]] or the [[Todd-Coxeter process]].

== Connection to graph theory ==
Insights into the structure of the group can be obtained by studying the [[adjacency matrix]] of the graph and in particular applying the theorems of [[spectral graph theory]].

A standard Cayley graph for the [[direct product]] of groups is the [[cartesian product]] of the corresponding Cayley graphs. For instance, a [[cycle graph|cycle]] <math>C_n</math> is a Cayley graph for the [[cyclic group]] <math>Z_n</math>. Therefore
the cartesian product <math> C_n \square C_m </math>, (an n by m grid on [[torus]]) is
a Cayley graph for the direct product <math> Z_n \times Z_m</math>.

== See also ==
* [[Vertex-transitive graph]]
* [[Generating set of a group]]
* [[Presentation of a group]]
* [[Lovász conjecture]]

[[Category:Group theory]]
[[Category:Permutation groups]]
[[Category:Graphs]]
[[Category:Geometric group theory]]
[[Category:Algebraic graph theory]]

Cayleygraph

2006-06-06T16:01:13Z

AugPi: another example

[[Image:Cayley graph of F2.svg|right|thumb|The Cayley graph of the [[free group]] on two generators ''a'' and ''b'']]

In [[mathematics]], a '''Cayley graph''', named after [[Arthur Cayley]], is a [[graph theory|graph]] that encodes the structure of a [[group (mathematics)|group]]. It is a central tool in [[combinatorial group theory|combinatorial]] and [[geometric group theory]].

Let <math>G</math> be a group, and let <math>S</math> be a [[generating set of a group|generating set]] for <math>G</math>. The Cayley graph of <math>G</math> with respect to <math>S</math> has a vertex for every element of <math>G</math>, with an edge from <math>g</math> to <math>gs</math> for all elements <math>g\in G</math> and <math>s\in S</math>.

== Example ==
For example, the Cayley graph of the [[free group]] on two generators <math>a</math> and <math>b</math> is depicted above and to the right. (Note that <math>e</math> represents the [[identity element]].) Travelling right along an edge represents multiplying on the right by <math>a</math>, while travelling up corresponds to multiplying by <math>b</math>. Since the free group has no [[relation (mathematics)|relation]]s, the graph has no cycles.

== Another example ==
[[Image:CayleyGraphOfDihedralGroupD4.PNG|300px|right|thumb|The Cayley graph of the dihedral group D4 on two generators α and β]]
The Cayley graph of the [[dihedral group]] ''D''4 on two generators α and β is depicted to the right. Red arrows represent left-multiplication by element α. Since element β is [[Cayley table|self-inverse]], the blue lines which represent left-multiplication by element β are undirected. Therefore the graph is mixed: it has eight vertices, eight arrows, and four edges.


== Variations ==
The above definition gives a connected, directed graph. There are a number of slight variations on the definition:

#Usually <math>S</math> is not allowed to contain the identity element <math>e</math>.
#If the set <math>S</math> doesn't generate the whole group, the Cayley graph isn't connected.
#In some contexts, left multiplication is used instead of right. That is, edges go from <math>g</math> to <math>sg</math>.
#In many contexts, the generating set is assumed to be ''symmetric'', meaning that <math>s^{-1}</math> is in <math>S</math> whenever <math>s</math> is. In this case, the graph is undirected.

== The Sabidussi Theorem ==
<math>G</math> [[group action|acts]] on itself by multiplication on the left. This action induces an action of <math>G</math> on its Cayley graph. Explicitly, an element <math>h</math> sends a vertex <math>g</math> to the vertex <math>hg</math>, and the edge <math>(g,gs)</math> to the edge <math>(hg,hgs)</math>. Since the action of <math>G</math> on itself is [[transitive]], any Cayley graph is [[vertex-transitive]]. The [[Sabidussi theorem]] gives a characterization of Cayley graphs:
Graph <math>X</math> is a Cayley graph if and only if the automorphism group of <math>X</math> contains a subgroup <math>G</math> acting regularly on the vertex set of <math>X</math>.

== Schreier coset graph ==
If one, instead, takes the vertices to be right cosets of a fixed subgroup <math>H</math>, one obtains a related construction, the [[Schreier coset graph]], which is at the basis of [[coset enumeration]] or the [[Todd-Coxeter process]].

== Connection to graph theory ==
Insights into the structure of the group can be obtained by studying the [[adjacency matrix]] of the graph and in particular applying the theorems of [[spectral graph theory]].

A standard Cayley graph for the [[direct product]] of groups is the [[cartesian product]] of the corresponding Cayley graphs. For instance, a [[cycle graph|cycle]] <math>C_n</math> is a Cayley graph for the [[cyclic group]] <math>Z_n</math>. Therefore
the cartesian product <math> C_n \square C_m </math>, (an n by m grid on [[torus]]) is
a Cayley graph for the direct product <math> Z_n \times Z_m</math>.

== See also ==
* [[Vertex-transitive graph]]
* [[Generating set of a group]]
* [[Presentation of a group]]
* [[Lovász conjecture]]

[[Category:Group theory]]
[[Category:Permutation groups]]
[[Category:Graphs]]
[[Category:Geometric group theory]]
[[Category:Algebraic graph theory]]