Entropic force

Last updated

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. [1] [2]

Contents

Mathematical formulation

In the canonical ensemble, the entropic force associated to a macrostate partition is given by [3]

where is the temperature, is the entropy associated to the macrostate , and is the present macrostate. [4]

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. [5]

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. [5]

Brownian motion

The entropic approach to Brownian movement was initially proposed by R. M. Neumann. [3] [6] Neumann derived the entropic force for a particle undergoing three-dimensional Brownian motion using the 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:

Polymers

A standard example of an entropic force is the elasticity of a freely jointed polymer 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 constrained Lagrangian 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]

Hydrophobic force

Water drops on the surface of grass Drops I.jpg
Water drops on the surface of grass

Another example of an entropic force is the 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 dissolved substance. Each water molecule is capable of

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.

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 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.

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. [11] Structures of water-soluble proteins typically have a core in which hydrophobic side chains are buried from water, which stabilizes the folded state. [12] 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, [12] [13] [14] although formation of hydrogen bonds within the protein also stabilizes protein structure. [15] [16]

Colloids

Entropic forces are important and widespread in the physics of colloids, [17] where they are responsible for the depletion force, and the ordering of hard particles, such as the crystallization of hard spheres, the isotropic-nematic transition in liquid crystal phases of hard rods, and the ordering of hard polyhedra. [17] [18] Because of this, entropic forces can be an important driver of self-assembly [17]

Entropic forces arise in colloidal systems due to the osmotic pressure 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 [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]

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. [22] 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. [23] An analogous example was found in the actin cytoskeleton. Here, the actin-bundling protein anillin drives actin contractility in cytokinetic rings. [24]

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, 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.

Gravity

In 2009, Erik Verlinde argued that gravity can be explained as an entropic force. [4] 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. [4]

Other forces

In the wake of the discussion started by Verlinde, entropic explanations for other fundamental forces have been suggested, [25] including Coulomb's law. [26] [27] The same approach was argued to explain dark matter, dark energy and Pioneer 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 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 [32] [33] in the context of Occam's razor principle.

See also

Related Research Articles

<span class="mw-page-title-main">Colloid</span> Mixture of an insoluble substance microscopically dispersed throughout another substance

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 the definition to include substances like aerosols and gels. The term colloidal suspension refers unambiguously to the overall mixture. A colloid has a dispersed phase and a continuous phase. The dispersed phase particles have a diameter of approximately 1 nanometre to 1 micrometre.

<span class="mw-page-title-main">Hydrophobe</span> Molecule or surface that has no attraction to water

In chemistry, hydrophobicity is the physical property of a molecule that is seemingly repelled from a mass of water. In contrast, hydrophiles are attracted to water.

<span class="mw-page-title-main">Solvation</span> Association of molecules of a solvent with molecules or ions of a solute

Solvation describes the interaction of a solvent with dissolved molecules. Both ionized and uncharged molecules interact strongly with a solvent, and the strength and nature of this interaction influence many properties of the solute, including solubility, reactivity, and color, as well as influencing the properties of the solvent such as its viscosity and density. If the attractive forces between the solvent and solute particles are greater than the attractive forces holding the solute particles together, the solvent particles pull the solute particles apart and surround them. The surrounded solute particles then move away from the solid solute and out into the solution. Ions are surrounded by a concentric shell of solvent. Solvation is the process of reorganizing solvent and solute molecules into solvation complexes and involves bond formation, hydrogen bonding, and van der Waals forces. Solvation of a solute by water is called hydration.

<span class="mw-page-title-main">Protein folding</span> Change of a linear protein chain to a 3D structure

Protein folding is the physical process by which a protein, after synthesis by a ribosome as a linear chain of amino acids, changes from an unstable random coil into a more ordered three-dimensional structure. This structure permits the protein to become biologically functional.

<span class="mw-page-title-main">Molecular dynamics</span> Computer simulations to discover and understand chemical properties

Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. The method is applied mostly in chemical physics, materials science, and biophysics.

<span class="mw-page-title-main">Optical tweezers</span> Scientific instruments

Optical tweezers are scientific instruments that use a highly focused laser beam to hold and move microscopic and sub-microscopic objects like atoms, nanoparticles and droplets, in a manner similar to tweezers. If the object is held in air or vacuum without additional support, it can be called optical levitation.

<span class="mw-page-title-main">Micelle</span> Group of fatty molecules suspended in liquid by soaps and/or detergents

A micelle or micella is an aggregate of surfactant amphipathic lipid molecules dispersed in a liquid, forming a colloidal suspension. A typical micelle in water forms an aggregate with the hydrophilic "head" regions in contact with surrounding solvent, sequestering the hydrophobic single-tail regions in the micelle centre.

<span class="mw-page-title-main">Self-assembly</span> Process in which disordered components form an organized structure or pattern

Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the constitutive components are molecules, the process is termed molecular self-assembly.

<span class="mw-page-title-main">Soft matter</span> Subfield of condensed matter physics

Soft matter or soft condensed matter is a subfield of condensed matter comprising a variety of physical systems that are deformed or structurally altered by thermal or mechanical stress of the magnitude of thermal fluctuations. These materials share an important common feature in that predominant physical behaviors occur at an energy scale comparable with room temperature thermal energy, and that entropy is considered the dominant factor. At these temperatures, quantum aspects are generally unimportant. Soft materials include liquids, colloids, polymers, foams, gels, granular materials, liquid crystals, flesh, and a number of biomaterials. When soft materials interact favorably with surfaces, they become squashed without an external compressive force. Pierre-Gilles de Gennes, who has been called the "founding father of soft matter," received the Nobel Prize in Physics in 1991 for discovering that methods developed for studying order phenomena in simple systems can be generalized to the more complex cases found in soft matter, in particular, to the behaviors of liquid crystals and polymers.

A chaotropic agent is a molecule in water solution that can disrupt the hydrogen bonding network between water molecules. This has an effect on the stability of the native state of other molecules in the solution, mainly macromolecules by weakening the hydrophobic effect. For example, a chaotropic agent reduces the amount of order in the structure of a protein formed by water molecules, both in the bulk and the hydration shells around hydrophobic amino acids, and may cause its denaturation.

<span class="mw-page-title-main">Hydrophobic effect</span> Aggregation of non-polar molecules in aqueous solutions

The hydrophobic effect is the observed tendency of nonpolar substances to aggregate in an aqueous solution and to be excluded by water. The word hydrophobic literally means "water-fearing", and it describes the segregation of water and nonpolar substances, which maximizes the entropy of water and minimizes the area of contact between water and nonpolar molecules. In terms of thermodynamics, the hydrophobic effect is the free energy change of water surrounding a solute. A positive free energy change of the surrounding solvent indicates hydrophobicity, whereas a negative free energy change implies hydrophilicity.

The DLVO theory explains the aggregation and kinetic stability of aqueous dispersions quantitatively and describes the force between charged surfaces interacting through a liquid medium. It combines the effects of the van der Waals attraction and the electrostatic repulsion due to the so-called double layer of counterions. The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials - that is when the potential energy of an elementary charge on the surface is much smaller than the thermal energy scale, . For two spheres of radius each having a charge separated by a center-to-center distance in a fluid of dielectric constant Failed to parse : {\displaystyle \epsilon_r} containing a concentration of monovalent ions, the electrostatic potential takes the form of a screened-Coulomb or Yukawa potential,

<span class="mw-page-title-main">Molecular motor</span> Biological molecular machines

Molecular motors are natural (biological) or artificial molecular machines that are the essential agents of movement in living organisms. In general terms, a motor is a device that consumes energy in one form and converts it into motion or mechanical work; for example, many protein-based molecular motors harness the chemical free energy released by the hydrolysis of ATP in order to perform mechanical work. In terms of energetic efficiency, this type of motor can be superior to currently available man-made motors. One important difference between molecular motors and macroscopic motors is that molecular motors operate in the thermal bath, an environment in which the fluctuations due to thermal noise are significant.

Poly(N-isopropylacrylamide) (variously abbreviated PNIPA, PNIPAM, PNIPAAm, NIPA, PNIPAA or PNIPAm) is a temperature-responsive polymer that was first synthesized in the 1950s. It can be synthesized from N-isopropylacrylamide which is commercially available. It is synthesized via free-radical polymerization and is readily functionalized making it useful in a variety of applications.

Hydrophobicity scales are values that define the relative hydrophobicity or hydrophilicity of amino acid residues. The more positive the value, the more hydrophobic are the amino acids located in that region of the protein. These scales are commonly used to predict the transmembrane alpha-helices of membrane proteins. When consecutively measuring amino acids of a protein, changes in value indicate attraction of specific protein regions towards the hydrophobic region inside lipid bilayer.

Adsorption of polyelectrolytes on solid substrates is a surface phenomenon where long-chained polymer molecules with charged groups bind to a surface that is charged in the opposite polarity. On the molecular level, the polymers do not actually bond to the surface, but tend to "stick" to the surface via intermolecular forces and the charges created by the dissociation of various side groups of the polymer. Because the polymer molecules are so long, they have a large amount of surface area with which to contact the surface and thus do not desorb as small molecules are likely to do. This means that adsorbed layers of polyelectrolytes form a very durable coating. Due to this important characteristic of polyelectrolyte layers they are used extensively in industry as flocculants, for solubilization, as supersorbers, antistatic agents, as oil recovery aids, as gelling aids in nutrition, additives in concrete, or for blood compatibility enhancement to name a few.

<span class="mw-page-title-main">Sharon Glotzer</span> American physicist

Sharon C. Glotzer is an American scientist and "digital alchemist", the Anthony C. Lembke Department Chair of Chemical Engineering, the John Werner Cahn Distinguished University Professor of Engineering and the Stuart W. Churchill Collegiate Professor of Chemical Engineering at the University of Michigan, where she is also professor of materials science and engineering, professor of physics, professor of macromolecular science and engineering, and professor of applied physics. She is recognized for her contributions to the fields of soft matter and computational science, most notably on problems in assembly science and engineering, nanoscience, and the glass transition, for which the elucidation of the nature of dynamical heterogeneity in glassy liquids is of particular significance. She is a member of the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences.

<span class="mw-page-title-main">Bovine submaxillary mucin coatings</span> Surface treatment for biomaterials

Bovine submaxillary mucin (BSM) coatings are a surface treatment provided to biomaterials intended to reduce the growth of disadvantageous bacteria and fungi such as S. epidermidis, E. coli, and Candida albicans. BSM is a substance extracted from the fresh salivary glands of cows. It exhibits unique physical properties, such as high molecular weight and amphiphilicity, that allow it to be used for many biomedical applications.

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 particles. One of the earliest reports of depletion forces that lead to particle coagulation is that of Bondy, who observed the separation or "creaming" of rubber latex upon addition of polymer depletant molecules to solution. More generally, depletants can include polymers, micelles, osmolytes, ink, mud, or paint dispersed in a continuous phase.

Patchy particles are micron- or nanoscale colloidal particles that are anisotropically patterned, either by modification of the particle surface chemistry, through particle shape, or both. The particles have a repulsive core and highly interactive surfaces that allow for this assembly. The placement of these patches on the surface of a particle promotes bonding with patches on other particles. Patchy particles are used as a shorthand for modelling anisotropic colloids, proteins and water and for designing approaches to nanoparticle synthesis. Patchy particles range in valency from two or higher. Patchy particles of valency three or more experience liquid-liquid phase separation. Some phase diagrams of patchy particles do not follow the law of rectilinear diameters.

References

  1. ↑ Müller, Ingo (2007). A History of Thermodynamics: The Doctrine of Energy and Entropy. Springer Science & Business Media. p. 115. ISBN   978-3-540-46227-9.
  2. ↑ Roos, Nico (2014). "Entropic forces in Brownian motion". American Journal of Physics. 82 (12): 1161–1166. arXiv: 1310.4139 . Bibcode:2014AmJPh..82.1161R. doi:10.1119/1.4894381. ISSN   0002-9505. S2CID   119286756.
  3. 1 2 Neumann RM (1980). "Entropic approach to Brownian movement". American Journal of Physics. 48 (5): 354–357. arXiv: 1310.4139 . Bibcode:1980AmJPh..48..354N. doi:10.1119/1.12095.
  4. 1 2 3 Verlinde, Erik (2011). "On the Origin of Gravity and the Laws of Newton". Journal of High Energy Physics. 2011 (4): 29. arXiv: 1001.0785 . Bibcode:2011JHEP...04..029V. doi:10.1007/JHEP04(2011)029. S2CID   3597565.
  5. 1 2 3 Taylor; Tabachnik (2013). "Entropic forces—making the connection between mechanics and thermodynamics in an exactly soluble model". European Journal of Physics. 34 (3): 729–736. Bibcode:2013EJPh...34..729T. doi:10.1088/0143-0807/34/3/729. S2CID   121469422.
  6. 1 2 Neumann RM (1977). "The entropy of a single Gaussian macromolecule in a noninteracting solvent". The Journal of Chemical Physics. 66 (2): 870–871. Bibcode:1977JChPh..66..870N. doi:10.1063/1.433923.
  7. ↑ Smith, SB; Finzi, L.; Bustamante, C. (1992). "Direct mechanical measurements of the elasticity of single DNA molecules by using magnetic beads". Science. 258 (5085): 1122–1126. Bibcode:1992Sci...258.1122S. doi:10.1126/science.1439819. PMID   1439819.
  8. ↑ Waters, James T.; Kim, Harold D. (April 18, 2016). "Force distribution in a semiflexible loop". Physical Review E. 93 (4): 043315. arXiv: 1602.08197 . Bibcode:2016PhRvE..93d3315W. doi:10.1103/PhysRevE.93.043315. PMC   5295765 . PMID   27176436.
  9. ↑ Keul ND, Oruganty K, Schaper Bergman ET, Beattie NR, McDonald WE, Kadirvelraj R, et al. (2018). "The entropic force generated by intrinsically disordered segments tunes protein function". Nature. 563 (7732): 584–588. Bibcode:2018Natur.563..584K. doi:10.1038/s41586-018-0699-5. PMC   6415545 . PMID   30420606.
  10. ↑ Encyclopedia of Life Science Article on Hydrophobic Effect (PDF). Figure 4. Archived from the original (PDF) on December 22, 2014. Retrieved April 10, 2012.
  11. ↑ "Essential Biochemistry".
  12. 1 2 Pace CN, Shirley BA, McNutt M, Gajiwala K (January 1, 1996). "Forces contributing to the conformational stability of proteins". FASEB J. 10 (1): 75–83. doi: 10.1096/fasebj.10.1.8566551 . PMID   8566551. S2CID   20021399.
  13. ↑ Compiani M, Capriotti E (December 2013). "Computational and theoretical methods for protein folding" (PDF). Biochemistry. 52 (48): 8601–8624. doi:10.1021/bi4001529. hdl:11585/564977. PMID   24187909. Archived from the original (PDF) on September 4, 2015.
  14. ↑ Callaway, David J. E. (1994). "Solvent-induced organization: a physical model of folding myoglobin". Proteins: Structure, Function, and Bioinformatics. 20 (1): 124–138. arXiv: cond-mat/9406071 . Bibcode:1994cond.mat..6071C. doi:10.1002/prot.340200203. PMID   7846023. S2CID   317080.
  15. ↑ Rose GD, Fleming PJ, Banavar JR, Maritan A (2006). "A backbone-based theory of protein folding". Proc. Natl. Acad. Sci. U.S.A. 103 (45): 16623–16633. Bibcode:2006PNAS..10316623R. doi: 10.1073/pnas.0606843103 . PMC   1636505 . PMID   17075053.
  16. ↑ Gerald Karp (2009). Cell and Molecular Biology: Concepts and Experiments. John Wiley and Sons. pp. 128–. ISBN   978-0-470-48337-4.
  17. 1 2 3 4 van Anders, Greg; Klotsa, Daphne; Ahmed, N. Khalid; Engel, Michael; Glotzer, Sharon C. (2014). "Understanding shape entropy through local dense packing". Proc Natl Acad Sci USA. 111 (45): E4812–E4821. arXiv: 1309.1187 . Bibcode:2014PNAS..111E4812V. doi: 10.1073/pnas.1418159111 . PMC   4234574 . PMID   25344532.
  18. 1 2 Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (2012). "Predictive Self-Assembly of Polyhedra into Complex Structures". Science. 337 (6093): 453–457. arXiv: 1202.2177 . Bibcode:2012Sci...337..453D. doi:10.1126/science.1220869. PMID   22837525. S2CID   7177740.
  19. ↑ van Anders, Greg; Ahmed, N. Khalid; Smith, Ross; Engel, Michael; Glotzer, Sharon C. (2014). "Entropically Patchy Particles: Engineering Valence through Shape Entropy". ACS Nano. 8 (1): 931–940. arXiv: 1304.7545 . doi:10.1021/nn4057353. PMID   24359081. S2CID   9669569.
  20. ↑ Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (2012). "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces". ACS Nano. 6 (1): 609–14. arXiv: 1109.1323 . doi:10.1021/nn204012y. PMID   22098586. S2CID   12785227.
  21. ↑ van Anders, Greg; Ahmed, N. Khalid; Smith, Ross; Engel, Michael; Glotzer, Sharon C. (2014). "Entropically Patchy Particles: Engineering Valence through Shape Entropy". ACS Nano. 8 (1): 931–940. arXiv: 1304.7545 . doi:10.1021/nn4057353. PMID   24359081. S2CID   9669569.
  22. ↑ Braun, Marcus; Lansky, Zdenek; Hilitski, Feodor; Dogic, Zvonimir; Diez, Stefan (2016). "Entropic forces drive contraction of cytoskeletal networks". BioEssays. 38 (5): 474–481. doi: 10.1002/bies.201500183 . PMID   26996935.
  23. ↑ Lansky, Zdenek; Braun, Marcus; Luedecke, Annemarie; Schlierf, Michael; ten Wolde, Pieter Rijn; Janson, Marcel E; Diez, Stefan (2015). "Diffusible Crosslinkers Generate Directed Forces in Microtubule Networks". Cell. 160 (6): 1159–1168. doi: 10.1016/j.cell.2015.01.051 . PMID   25748652. S2CID   14647448.
  24. ↑ Kucera, Ondrej; Siahaan, Valerie; Janda, Daniel; Dijkstra, Sietske H; Pilatova, Eliska; Zatecka, Eva; Diez, Stefan; Braun, Marcus; Lansky, Zdenek (2021). "Anillin propels myosin-independent constriction of actin rings". Nature Communications. 12 (1): 4595. Bibcode:2021NatCo..12.4595K. doi:10.1038/s41467-021-24474-1. PMC   8319318 . PMID   34321459.
  25. 1 2 Visser, Matt (2011). "Conservative entropic forces". Journal of High Energy Physics. 2011 (10): 140. arXiv: 1108.5240 . Bibcode:2011JHEP...10..140V. doi:10.1007/JHEP10(2011)140. S2CID   119097091.
  26. ↑ Wang, Tower (2010). "Coulomb force as an entropic force". Physical Review D. 81 (10): 104045. arXiv: 1001.4965 . Bibcode:2010PhRvD..81j4045W. doi:10.1103/PhysRevD.81.104045. S2CID   118545831.
  27. ↑ Hendi, S. H.; Sheykhi, A. (2012). "Entropic Corrections to Coulomb's Law". International Journal of Theoretical Physics. 51 (4): 1125–1136. arXiv: 1009.5561 . Bibcode:2012IJTP...51.1125H. doi:10.1007/s10773-011-0989-2. S2CID   118849945.
  28. ↑ Chang, Zhe; Li, Ming-Hua; Li, Xin (2011). "Unification of Dark Matter and Dark Energy in a Modified Entropic Force Model". Communications in Theoretical Physics. 56 (1): 184–192. arXiv: 1009.1506 . Bibcode:2011CoTPh..56..184C. doi:10.1088/0253-6102/56/1/32. S2CID   119312663.
  29. ↑ Wissner-Gross, A.D.; Freer, C.E. (2013). "Causal Entropic Forces" (PDF). Physical Review Letters. 110 (16): 168702. Bibcode:2013PhRvL.110p8702W. doi: 10.1103/PhysRevLett.110.168702 . PMID   23679649.
  30. ↑ Canessa, E. (2013). "Comment on Phys. Rev. Lett. 110, 168702 (2013): Causal Entropic Forces". arXiv: 1308.4375 [cond-mat.dis-nn].
  31. ↑ Kappen, H. J. (2013). "Comment: Causal entropic forces". arXiv: 1312.4185 [cond-mat.stat-mech].
  32. ↑ Soklakov, Andrei N. (2000). "Occam's Razor as a Formal Basis for a Physical Theory". arXiv: math-ph/0009007 .
  33. ↑ Soklakov, Andrei N. (2000). "Complexity analysis for algorithmically simple strings". arXiv: cs/0009001 .
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.