Inphysics, anentropic force acting in a system is anemergent phenomenon resulting from the entire system's statistical tendency to increase itsentropy, rather than from a particular underlyingforce on the atomic scale.^[1][2]

In thecanonical ensemble, the entropic force${\displaystyle \mathbf{F}}$ associated to a macrostate partition${\displaystyle \{\mathbf{X}\}}$ is given by^[3]

${\displaystyle \mathbf{F}({\mathbf{X}}_0)=T{\mathrm{\nabla }}_{\mathbf{X}}S(\mathbf{X}){|}_{{\mathbf{X}}_0},}$

where${\displaystyle T}$ is the temperature,${\displaystyle S(\mathbf{X})}$ is the entropy associated to the macrostate${\displaystyle \mathbf{X}}$, and${\displaystyle {\mathbf{X}}_0}$ is the present macrostate.^[4]

Theinternal energy of anideal gas depends only on its temperature, and not on the volume of its containing box, so it is not anenergy effect that tends to increase the volume of the box as gaspressure does. This implies that thepressure of an ideal gas has an entropic origin.^[5]

What is the origin of such an entropic force? The most general answer is that the effect of thermal fluctuations tends to bring athermodynamic system toward a macroscopic state that corresponds to a maximum in the number ofmicroscopic 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 maximumentropy.^[5]

The entropic approach toBrownian movement was initially proposed by R. M. Neumann.^[3][6] Neumann derived the entropic force for a particle undergoing three-dimensional Brownian motion using theBoltzmann equation, denoting this force as adiffusional driving force orradial force. In the paper, three example systems are shown to exhibit such a force:

• electrostatic system ofmolten salt,
• surface tension and,
• elasticity of rubber.

A standard example of an entropic force is theelasticity of a freely jointedpolymer molecule.^[6] 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.^[5][7] The entropic force by a freely jointed chain has a clear mechanical origin and can be computed using constrainedLagrangian dynamics.^[8] 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.^[9]

Another example of an entropic force is thehydrophobic force. At room temperature, it partly originates from the loss of entropy by the 3D network of water molecules when they interact with molecules ofdissolved substance. Each water molecule is capable of

• donating twohydrogen bonds through the two protons,
• accepting two more hydrogen bonds through the twosp³-hybridizedlone pairs.

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 tohydrogen fluoride (which can accept 3 but donate only 1) orammonia (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.^[10] 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 aclathrate-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 isprotein folding, which is aspontaneous process and wherehydrophobic effect also plays a role.^[11] Structures of water-soluble proteins typically have a core in which hydrophobicside chains are buried from water, which stabilizes the folded state.^[12] Charged andpolar 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,^[12][13][14] although formation of hydrogen bonds within the protein also stabilizes protein structure.^[15][16]

Entropic forces are important and widespread in the physics ofcolloids,^[17] where they are responsible for thedepletion force, and the ordering of hard particles, such as thecrystallization ofhard spheres, the isotropic-nematic transition inliquid crystal phases of hard rods, and the ordering of hard polyhedra.^[17][18] Because of this, entropic forces can be an important driver ofself-assembly^[17]

Entropic forces arise in colloidal systems due to theosmotic pressure that comes from particle crowding. This was first discovered in, and is most intuitive for, colloid-polymer mixtures described by theAsakura–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^[17] of colloids into a diverse array of structures^[18] that can be rationally designed by modifying the shape of the particles.^[19] These effects are for anisotropic particles referred to as directional entropic forces.^[20][21]

Contractile forces in biological cells are typically driven bymolecular motors associated with thecytoskeleton. However, a growing body of evidence shows that contractile forces may also be of entropic origin.^[22] The foundational example is the action of microtubule crosslinker Ase1, which localizes tomicrotubule overlaps in themitotic 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.^[23] An analogous example was found in theactin cytoskeleton. Here, the actin-bundling proteinanillin drives actin contractility in cytokinetic rings.^[24]

Some forces that are generally regarded asconventional forces have been argued to be actually entropic in nature. These theories remain controversial and are the subject of ongoing work.Matt Visser, professor of mathematics at Victoria University of Wellington, NZ in "Conservative Entropic Forces"^[25] criticizes selected approaches but generally concludes:

There is no reasonable doubt concerning the physical reality of entropic forces, and no reasonable doubt that classical (and semi-classical) general relativity is closely related to thermodynamics. Based on the work of Jacobson,Thanu Padmanabhan, and others, there are also good reasons to suspect a thermodynamic interpretation of the fully relativistic Einstein equations might be possible.

In 2009,Erik Verlinde argued that gravity can be explained as an entropic force.^[4] It claimed (similar toJacobson'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 withGerard 't Hooft'sholographic principle. It implies that gravity is not afundamental interaction, but anemergent phenomenon.^[4]

In the wake of the discussion started by Verlinde, entropic explanations for other fundamental forces have been suggested,^[25] includingCoulomb's law.^[26][27] The same approach was argued to explaindark matter,dark energy andPioneer effect.^[28]

It was argued that causal entropic forces lead to spontaneous emergence of tool use and social cooperation.^[29][30][31] Causal entropic forces by definition maximizeentropy 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^[32][33] in the context ofOccam's razor principle.

• Materials science
• Thermodynamic entropy
• Soft matter

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