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| [[File:DiffusionMicroMacro.gif|thumb|250px|Diffusion from a microscopic and macroscopic point of view. Initially, there are [[solution|solute]] molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. <u>Top:</u> A single molecule moves around randomly. <u>Middle:</u> With more molecules, there is a statistical trend that the solute fills the container more and more uniformly. <u>Bottom:</u> With an enormous number of solute molecules, all randomness is gone: The solute appears to move smoothly and deterministically from high-concentration areas to low-concentration areas. There is no microscopic [[force]] pushing molecules rightward, but there ''appears'' to be one in the bottom panel. This fake-force is called an '''entropic force'''.]]
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| In [[physics]], an '''entropic force''' acting in a system is a [[phenomenological force]] resulting from the entire system's statistical tendency to increase its [[entropy]], rather than from a particular underlying [[microscopic]] [[force]].<ref>[http://books.google.com/books?id=u13KiGlz2zcC&pg=PA115 ''A history of thermodynamics: the doctrine of energy and entropy'' by Ingo Müller, p115]</ref>
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| ==Mathematical formulation==
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| In the [[canonical ensemble]], the entropic force <math>\mathbf F</math> associated to a macrostate partition <math>\{\mathbf{X}\}</math> is given by: <ref>Entropic approach to Brownian movement; See Richard M. Neumann, Am. J. Phys. 48, 354 (1980) http://dx.doi.org/10.1119/1.12095</ref> <ref name="cef">[http://www.alexwg.org/link?url=http%3A%2F%2Fwww.alexwg.org%2Fpublications%2FPhysRevLett_110-168702.pdf Causal entropic forces], [[Alex Wissner-Gross]], ''Physical Review Letters 110, 168702 (2013)''</ref><ref>[http://link.springer.com/content/pdf/10.1007%2FJHEP04%282011%29029.pdf On the origin of gravity and the laws of Newton], [[Erik Verlinde]]</ref>
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| <center>
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| <math>\mathbf{F}(\mathbf{X_0}) = T \nabla_{\mathbf X} S(\mathbf{X})|_{\mathbf X_0}</math>
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| </center>
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| where <math>T</math> is the temperature, <math>S(\mathbf{X})</math> is the entropy associated to the macrostate <math>\mathbf{X}</math> and <math>\mathbf{X_0}</math> is the present macrostate.
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| ==Examples==
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| === Brownian Motion ===
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| Entropic approach to [[Brownian motion|Brownian movement]] is initially proposed by R. M. Neumann,<ref>Entropic approach to Brownian movement; See Richard M. Neumann, Am. J. Phys. 48, 354 (1980) http://dx.doi.org/10.1119/1.12095</ref>
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| .<ref name="dx.doi.org">The entropy of a single Gaussian macromolecule in a noninteracting solvent; See Richard M. Neumann, J. Chem. Phys. 66 , 870 (1977) ; http://dx.doi.org/10.1063/1.433923</ref> Neumann derived the entropic force for a particle undergoing three-dimensional Brownian motion using [[Boltzmann equation]], denoting this force as a ''diffusional driving force'' or ''radial force''. In the paper, three example systems are shown to exhibit such a force [[Coulomb's law|electrostatic system]] of [[molten salt]], [[surface tension]] and rubber elasticity.
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| ===Polymers===
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| A standard example of an entropic force is the [[Elasticity (physics)|elasticity]] of a freely-jointed [[polymer]] molecule described by a [[Gaussian distribution]].<ref name="dx.doi.org"/> If the molecule is pulled into an extended configuration, the system has an increased amount of predictability. But randomly coiled configurations are overwhelmingly more probable; i.e., they have greater entropy. This results in the chain eventually returning (through [[diffusion]]) to such a configuration. To the macroscopic observer, the precise origin of the microscopic forces that drive the motion is irrelevant. The observer simply sees the polymer contract into a state of higher entropy, as if driven by an elastic force.
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| ===Hydrophobic force===
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| {{See also|Hydrophobic effect#The origin of hydrophobic effect}}
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| [[Image:Drops I.jpg|thumb|right|100px|Water drops on the surface of grass.]]Another example of an entropic force is the [[hydrophobe|hydrophobic]] force. At room temperature, it partly originates from the loss of entropy by the 3D network of water molecules when they interact with molecules of [[solute|dissolved substance]]. Each water molecule is capable of
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| * donating two [[hydrogen bond]]s through the two protons
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| * accepting two more hydrogen bonds through the two [[sp3 hybridization|sp<sup>3</sup>-hybridized]] [[lone pairs]]
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| Therefore, water molecules can form an extended three-dimensional network. Introduction of a non-hydrogen-bonding surface disrupts this network. The water molecules rearrange themselves around the surface, so as to minimize the number of disrupted hydrogen bonds. This is in contrast to [[hydrogen fluoride]] (which can accept 3 but donate only 1) or [[ammonia]] (which can donate 3 but accept only 1), which mainly form linear chains.
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| If the introduced surface had an ionic or polar nature, there would be water molecules standing upright on 1 (along the axis of an orbital for ionic bond) or 2 (along a resultant polarity axis) of the four sp<sup>3</sup> orbitals.<ref>Encyclopedia of Life Science Article on Hydrophobic Effect; See Figure 4: http://xibalba.lcg.unam.mx/~rgalindo/bioquimica/BQPosgrado2011/I%20FQ%20repaso/HydrophobicEffect.pdf</ref> These orientations allow easy movement, i.e. degrees of freedom, and thus lowers entropy minimally. But a non-hydrogen-bonding surface with a moderate curvature forces the water molecule to sit tight on the surface, spreading 3 hydrogen bonds tangential to the surface, which then become locked in a [[clathrate]]-like basket shape. Water molecules involved in this clathrate-like basket around the non-hydrogen-bonding surface are constrained in their orientation. Thus, any event that would minimize such a surface is entropically favored. For example, when two such hydrophobic particles come very close, the clathrate-like baskets surrounding them merge. This releases some of the water molecules into the bulk of the water, leading to an increase in entropy. This is the basis of the so-called "attraction" between hydrophobic objects in water.
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| ===Directional Entropic Force===
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| Entropic forces also occur in the physics of gases and solutions, where they generate the pressure of an [[ideal gas]] (the energy of which depends only on its [[temperature]], not its volume), the [[osmotic pressure]] of a dilute solution, and in [[colloids|colloidal suspensions]], where they are responsible for the [[crystallization]] of hard spheres.
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| In nano and colloidal science, Entropic Forces usually come from the effect of depletion, where small particles induce crystallization of bigger ones.
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| Even in the absence of depletion, however, scientist [[Sharon Glotzer]] and collaborators recently conjectured that '''Directional Entropic Forces''' could be responsible for the alignment of facets observed prior the assembly and/or crystallization of systems of polyhedral nano and colloidal particles.<ref>{{cite web |url=http://pubs.acs.org/doi/abs/10.1021/nn204012y |title=Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces |accessdate=23 June 2012 |publisher=[[American Chemical Society|ACS]]|archiveurl=http://arxiv.org/pdf/1109.1323v2.pdf|archivedate = 2011-12-01}}</ref> This was later proven to be correct<ref>{{cite web |url=http://arxiv.org/pdf/1309.1187.pdf |title=Unified Theoretical Framework for Shape Entropy in Colloids |accessdate=20 October 2013 |archiveurl=http://arxiv.org/pdf/1309.1187.pdf |archivedate = 2013-09-03}}</ref><ref>{{cite web |url=http://www-personal.umich.edu/~damascus/files/Angew2013.pdf |title=A Directional Entropic Force Approach to Assemble Anisotropic Nanoparticles into Superlattices |accessdate=13 Jan 2014 |archiveurl=http://www-personal.umich.edu/~damascus/files/Angew2013.pdf |archivedate = 2013-09-03}}</ref> and allowed for the development of a roadmap for the assembly of polyhedral particles into atomic isostructures.<ref>{{cite web |url=http://www-personal.umich.edu/~damascus/files/Science-2012-Damasceno.pdf |title=Structural Diversity and the Role of Particle Shape and Dense Fluid Behavior in Assemblies of Hard Polyhedra |accessdate=23 June 2012 |archiveurl=http://arxiv.org/pdf/1202.2177v1.pdf|archivedate = 2012-02-10}}</ref>
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| ==Speculative examples==
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| In recent years (especially since 2009) {{citation needed|date=June 2013}} some forces that are generally regarded as [[force|conventional forces]] have been argued to be actually entropic in nature. These theories remain speculative and are the subject of ongoing work.
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| ===Gravity===
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| {{main|Entropic gravity}}
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| It is generally believed that gravity is a microscopic [[force]]{{citation needed|date=June 2012}} (or arguably a [[pseudo-force]] in [[general relativity]] where it becomes a [[pseudo-scalar]] at the [[Planck]] length scale). However, in 2009, [[Erik Verlinde]]<ref name="VerlindePaper">{{cite arXiv|last=Verlinde|first=Eric|title=Title: On the Origin of Gravity and the Laws of Newton|eprint=1001.0785|date=6 January 2010|class=hep-th}}</ref> argued that gravity can be explained as an entropic force.<ref>{{cite journal |author=E.P. Verlinde |doi=10.1007/JHEP04(2011)029 |journal=JHEP 04, 29 (2011) |title=On the Origin of Gravity and the Laws of Newton |bibcode = 2011JHEP...04..029V |arxiv = 1001.0785 }}</ref>
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| For example, when someone throws a ball in the air, it follows a [[parabolic trajectory]] (in the absence of wind resistance). Conventionally, it is said that the ball follows a deterministic path dictated by [[Newton's law of gravity]] or [[general relativity]]. However, in the entropic theory, it is argued that the ball can follow any trajectory and picks a trajectory "at random."{{clarify|date=June 2012}} A calculation demonstrates that, in the collection of possible trajectories, the overwhelming majority are almost exactly the same as the parabolic trajectory; therefore, the ball is observed to follow a parabola.
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| ===Intelligence===
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| In 2013, [[Alex Wissner-Gross]] argued that some intelligent-seeming behavior could be modelled as a causal entropic force.<ref name="cef" />
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| ===Other forces===
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| Other fundamental forces have been argued recently to be entropic in origin, including [[Coulomb's law]],<ref>http://arxiv.org//abs/1001.4965, ''Coulomb Force as an Entropic Force'', T. Wang</ref><ref>http://arxiv.org//abs/0809.4631, ''Simple field theoretical approach of Coulomb systems. Entropic effects'', D. di Caprio, J.P. Badiali, M. Holovko</ref><ref>http://arxiv.org//abs/1009.5561, ''Entropic Corrections to Coulomb's Law, '' A. Sheykhi, S. H. Hendi</ref> the [[electroweak]] and [[strong force]]s,<ref>http://arxiv.org//abs/1008.4147,'' Emergent Gauge Fields'', Peter G.O. Freund</ref> and [[dark matter]] and [[dark energy]].<ref>http://arxiv.org//abs/1009.1506 ''Unification of Dark Matter and Dark Energy in a Modified Entropic Force Model'', Zhe Chang, Ming-Hua Li, Xin Li</ref>
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| ===Links to Occam's Razor===
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| A formal simultaneous connection between the mathematical structure of the discovered laws of nature, intelligence and the entropy-like measures of complexity was previously noted in 2000 by Andrei Soklakov<ref>Andrei N. Soklakov, "Occam's razor as a formal basis for a physical theory" (arXiv:math-ph/0009007, September 2000; Foundations of Physics Letters, 2002), "Complexity analysis for algorithmically simple strings" (arXiv:cs/0009001, September 2000).</ref> in the context of Occam's razor principle.
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| ==See also==
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| * [[Colloid]]s
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| * [[Nanomechanics]]
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| * [[Data clustering]]
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| * [[Ideal chain#Entropic elasticity of an ideal chain|Entropic elasticity of an ideal chain]]
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| ==References==
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| <references/>
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| {{Chemical bonds}}
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| {{DEFAULTSORT:Entropic Force}}
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| [[Category:Materials science]]
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| [[Category:Thermodynamic entropy]]
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