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In physics, an entropic force acting in a system is an emergent phenomenon resulting from the entire system's statistical tendency to increase its entropy, rather than from a particular underlying force on the atomic scale.


Mathematical formulation

In the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
, the entropic force \mathbf F associated to a macrostate partition \ is given by: \mathbf(\mathbf) = T \nabla_ S(\mathbf), _ where T is the temperature, S(\mathbf) is the entropy associated to the macrostate \mathbf and \mathbf is the present macrostate.


Examples


Pressure of an ideal gas

The internal energy of an ideal gas depends only on its temperature, and not on the volume of its containing box, so it is not an energy effect that tends to increase the volume of the box as gas pressure does. This implies that the pressure of an ideal gas has an entropic origin. What is the origin of such an entropic force? The most general answer is that the effect of thermal fluctuations tends to bring a thermodynamic system toward a macroscopic state that corresponds to a maximum in the number of microscopic states (or micro-states) that are compatible with this macroscopic state. In other words, thermal fluctuations tend to bring a system toward its macroscopic state of maximum entropy.


Brownian motion

The entropic approach to Brownian movement was initially proposed by R. M. Neumann. Neumann derived the entropic force for a particle undergoing three-dimensional Brownian motion using the
Boltzmann equation The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
, denoting this force as a ''diffusional driving force'' or ''radial force''. In the paper, three example systems are shown to exhibit such a force: *
electrostatic system Electrostatics is a branch of physics that studies electric charges at Rest (physics), rest (static electricity). Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles af ...
of
molten salt Molten salt is salt which is solid at standard temperature and pressure but enters the liquid phase due to elevated temperature. Regular table salt has a melting point of 801 °C (1474°F) and a heat of fusion of 520 J/g.Journal of Chemical Th ...
*
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to ...
and * Elasticity of rubber.


Polymers

A standard example of an entropic force is the elasticity of a freely-jointed polymer molecule. For an ideal chain, maximizing its entropy means reducing the distance between its two free ends. Consequently, a force that tends to collapse the chain is exerted by the ideal chain between its two free ends. This entropic force is proportional to the distance between the two ends. The entropic force by a freely-jointed chain has a clear mechanical origin, and can be computed using constrained Lagrangian dynamics. With regards to biological polymers, there appears to be an intricate link between the entropic force and function. For example, disordered polypeptide segments – in the context of the folded regions of the same polypeptide chain – have been shown to generate an entropic force that has functional implications.


Hydrophobic force

Another example of an entropic force is the
hydrophobic In chemistry, hydrophobicity is the physical property of a molecule that is seemingly repelled from a mass of water (known as a hydrophobe). In contrast, hydrophiles are attracted to water. Hydrophobic molecules tend to be nonpolar and, t ...
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 dissolved substance. Each water molecule is capable of * donating two hydrogen bonds through the two protons * accepting two more hydrogen bonds through the two sp3-hybridized
lone pair In chemistry, a lone pair refers to a pair of valence electrons that are not shared with another atom in a covalent bondIUPAC ''Gold Book'' definition''lone (electron) pair''/ref> and is sometimes called an unshared pair or non-bonding pair. Lone ...
s 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 Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous wa ...
(which can donate 3 but accept only 1), which mainly form linear chains. 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 sp3 orbitals. 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 A clathrate is a chemical substance consisting of a lattice that traps or contains molecules. The word ''clathrate'' is derived from the Latin (), meaning ‘with bars, latticed’. Most clathrate compounds are polymeric and completely envelop ...
-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. Another related and counter-intuitive example of entropic force is protein folding, which is a spontaneous process and where hydrophobic effect also plays a role. Structures of water-soluble proteins typically have a core in which hydrophobic
side chains In organic chemistry and biochemistry, a side chain is a chemical group that is attached to a core part of the molecule called the "main chain" or backbone. The side chain is a hydrocarbon branching element of a molecule that is attached to a ...
are buried from water, which stabilizes the folded state. Charged and polar side chains are situated on the solvent-exposed surface where they interact with surrounding water molecules. Minimizing the number of hydrophobic side chains exposed to water is the principal driving force behind the folding process, although formation of hydrogen bonds within the protein also stabilizes protein structure.


Colloids

Entropic forces are important and widespread in the physics of
colloid A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend ...
s, where they are responsible for the
depletion force A depletion force is an effective attractive force that arises between large colloidal particles that are suspended in a dilute solution of ''depletants'', which are smaller solutes that are preferentially excluded from the vicinity of the large p ...
, and the ordering of hard particles, such as the crystallization of hard spheres, the isotropic-
nematic Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. The ...
transition in liquid crystal phases of hard rods, and the ordering of hard polyhedra. Because of this, entropic forces can be an important driver of self-assembly Entropic forces arise in colloidal systems due to the
osmotic pressure Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane. It is also defined as the measure of the tendency of a solution to take in a pure ...
that comes from particle crowding. This was first discovered in, and is most intuitive for, colloid-polymer mixtures described by the Asakura–Oosawa model. In this model, polymers are approximated as finite-sized spheres that can penetrate one another, but cannot penetrate the colloidal particles. The inability of the polymers to penetrate the colloids leads to a region around the colloids in which the polymer density is reduced. If the regions of reduced polymer density around two colloids overlap with one another, by means of the colloids approaching one another, the polymers in the system gain an additional free volume that is equal to the volume of the intersection of the reduced density regions. The additional free volume causes an increase in the entropy of the polymers, and drives them to form locally dense-packed aggregates. A similar effect occurs in sufficiently dense colloidal systems without polymers, where osmotic pressure also drives the local dense packing of colloids into a diverse array of structures that can be rationally designed by modifying the shape of the particles. These effects are for anisotropic particles referred to as directional entropic forces.


Cytoskeleton

Contractile forces in biological cells are typically driven by molecular motors associated with the cytoskeleton. However, a growing body of evidence shows that contractile forces may also be of entropic origin. The foundational example is the action of microtubule crosslinker Ase1, which localizes to microtubule overlaps in the mitotic spindle. Molecules of Ase1 are confined to the microtubule overlap, where they are free to diffuse one-dimensionally. Analogically to an ideal gas in a container, molecules of Ase1 generate pressure on the overlap ends. This pressure drives the overlap expansion, which results in the contractile sliding of the microtubules. An analogous example was found in the
actin Actin is a family of globular multi-functional proteins that form microfilaments in the cytoskeleton, and the thin filaments in muscle fibrils. It is found in essentially all eukaryotic cells, where it may be present at a concentration of ov ...
cytoskeleton. Here, the actin-bundling protein
anillin Anillin is a conserved protein implicated in cytoskeletal dynamics during cellularization and cytokinesis. The ''ANLN'' gene in humans and the scraps gene in ''Drosophila'' encode Anillin. In 1989, anillin was first isolated in embryos of ''Dros ...
drives actin contractility in cytokinetic rings.


Controversial examples

Some forces that are generally regarded as conventional forces have been argued to be actually entropic in nature. These theories remain controversial and are the subject of ongoing work.
Matt Visser Matt Visser is a mathematics Professor at Victoria University of Wellington, in New Zealand. Work Visser's research interests include general relativity, quantum field theory and cosmology. Visser has produced a large number of research paper ...
, professor of mathematics at Victoria University of Wellington, NZ in "Conservative Entropic Forces" criticizes selected approaches but generally concludes:


Gravity

In 2009, Erik Verlinde argued that gravity can be explained as an entropic force. It claimed (similar to Jacobson's result) that gravity is a consequence of the "information associated with the positions of material bodies". This model combines the thermodynamic approach to gravity with Gerard 't Hooft's holographic principle. It implies that gravity is not a fundamental interaction, but an emergent phenomenon.


Other forces

In the wake of the discussion started by Verlinde, entropic explanations for other fundamental forces have been suggested, including
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
. The same approach was argued to explain
dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
,
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...
and Pioneer effect.


Links to adaptive behavior

It was argued that causal entropic forces lead to spontaneous emergence of tool use and social cooperation. Causal entropic forces by definition maximize entropy production between the present and future time horizon, rather than just greedily maximizing instantaneous entropy production like typical entropic forces. 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 in the context of Occam's razor principle.


See also

*
Colloid A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend ...
s * Nanomechanics * Thermodynamics *
Abraham–Lorentz force In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation by self-interaction. It is also called ...
* Entropic gravity * Entropy *
Introduction to entropy In thermodynamics, entropy is a numerical quantity that shows that many physical processes can go in only one direction in time. For example, you can pour cream into coffee and mix it, but you cannot "unmix" it; you can burn a piece of wood, but ...
* Entropic elasticity of an ideal chain *
Hawking radiation Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
* Data clustering *
Depletion force A depletion force is an effective attractive force that arises between large colloidal particles that are suspended in a dilute solution of ''depletants'', which are smaller solutes that are preferentially excluded from the vicinity of the large p ...
* Maximal entropy random walk


References

{{Reflist, 30em Materials science Thermodynamic entropy Soft matter