Absolute value

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Absolute_value

The graph of the absolute value function for real numbers

The absolute value of a number may be thought of as its distance from zero.

In mathematics, the absolute value or modulus of a real number ${\displaystyle x}$, denoted ${\displaystyle |x|}$, is the non-negative value of ${\displaystyle x}$ without regard to its sign. Namely, ${\displaystyle |x|=x}$ if ${\displaystyle x}$ is a positive number, and ${\displaystyle |x|=-x}$ if ${\displaystyle x}$ is negative (in which case negating ${\displaystyle x}$ makes ${\displaystyle -x}$ positive), and ${\displaystyle |0|=0}$. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

Terminology and notation

In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specifically for the complex absolute value and it was borrowed into English in 1866 as the Latin equivalent modulus. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English. The notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude. The absolute value of ${\displaystyle x}$ has also been denoted ${\displaystyle \mathrm{abs}x}$ in some mathematical publications, and in spreadsheets, programming languages, and computational software packages, the absolute value of $x$ is generally represented by abs(x), or a similar expression, as it has been since the earliest days of high-level programming languages.

The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant. Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the Euclidean norm or sup norm of a vector in ${\displaystyle {\mathbb{R}}^n}$, although double vertical bars with subscripts (${\displaystyle \Vert \cdot {\Vert }_2}$ and ${\displaystyle \Vert \cdot {\Vert }_{\mathrm{\infty }}}$, respectively) are a more common and less ambiguous notation.

Definition and properties

Real numbers

For any real number ${\displaystyle x}$, the absolute value or modulus of ${\displaystyle x}$ is denoted by ${\displaystyle |x|}$, with a vertical bar on each side of the quantity, and is defined as $${\displaystyle |x|=\{\begin{array}{ll}x,& \text{if }x\ge 0\\ -x,& \text{if }x<0.\end{array}}$$

The absolute value of ${\displaystyle x}$ is thus always either a positive number or zero, but never negative. When ${\displaystyle x}$ itself is negative (${\displaystyle x<0}$), then its absolute value is necessarily positive (${\displaystyle |x|=-x>0}$).

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers (their absolute difference) is the distance between them. The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below).

Since the square root symbol represents the unique positive square root, when applied to a positive number, it follows that $${\displaystyle |x|=\sqrt{x^2}.}$$ This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.

The absolute value has the following four fundamental properties ($a$, $b$ are real numbers), that are used for generalization of this notion to other domains:

${\displaystyle |a|\ge 0}$	Non-negativity
${\displaystyle |a|=0⟺a=0}$	Positive-definiteness
${\displaystyle |ab|=\left|a\right|\left|b\right|}$	Multiplicativity
${\displaystyle |a+b|\le |a|+|b|}$	Subadditivity, specifically the triangle inequality

Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that ${\displaystyle |a+b|=s(a+b)}$ where ${\displaystyle s=\pm 1}$, with its sign chosen to make the result positive. Now, since ${\displaystyle -1\cdot x\le |x|}$ and ${\displaystyle +1\cdot x\le |x|}$, it follows that, whichever of ${\displaystyle \pm 1}$ is the value of ${\displaystyle s}$, one has ${\displaystyle s\cdot x\le |x|}$ for all real ${\displaystyle x}$. Consequently, ${\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\le |a|+|b|}$, as desired.

Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.

${\displaystyle |\left|a\right||=|a|}$	Idempotence (the absolute value of the absolute value is the absolute value)
${\displaystyle \left|-a\right|=|a|}$	Evenness (reflection symmetry of the graph)
${\displaystyle |a-b|=0⟺a=b}$	Identity of indiscernibles (equivalent to positive-definiteness)
${\displaystyle |a-b|\le |a-c|+|c-b|}$	Triangle inequality (equivalent to subadditivity)
${\displaystyle \left|\frac{a}{b}\right|=\frac{|a|}{|b|}}$ (if ${\displaystyle b\ne 0}$)	Preservation of division (equivalent to multiplicativity)
${\displaystyle |a-b|\ge |\left|a\right|-\left|b\right||}$	Reverse triangle inequality (equivalent to subadditivity)

Two other useful properties concerning inequalities are:

${\displaystyle |a|\le b⟺-b\le a\le b}$

${\displaystyle |a|\ge b⟺a\le -b}$ or ${\displaystyle a\ge b}$

These relations may be used to solve inequalities involving absolute values. For example:

${\displaystyle |x-3|\le 9}$	${\displaystyle ⟺-9\le x-3\le 9}$
	${\displaystyle ⟺-6\le x\le 12}$

The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers.

Complex numbers

The absolute value of a complex number ${\displaystyle z}$ is the distance ${\displaystyle r}$ of ${\displaystyle z}$ from the origin. It is also seen in the picture that ${\displaystyle z}$ and its complex conjugate ${\displaystyle \overline{z}}$ have the same absolute value.

Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number $${\displaystyle z=x+iy,}$$ where ${\displaystyle x}$ and ${\displaystyle y}$ are real numbers, the absolute value or modulus of ${\displaystyle z}$ is denoted ${\displaystyle |z|}$ and is defined by $${\displaystyle |z|=\sqrt{\mathrm{Re}(z{)}^2+\mathrm{Im}(z{)}^2}=\sqrt{x^2+y^2},}$$ the Pythagorean addition of ${\displaystyle x}$ and ${\displaystyle y}$, where ${\displaystyle \mathrm{Re}(z)=x}$ and ${\displaystyle \mathrm{Im}(z)=y}$ denote the real and imaginary parts of ${\displaystyle z}$, respectively. When the imaginary part ${\displaystyle y}$ is zero, this coincides with the definition of the absolute value of the real number ${\displaystyle x}$.

When a complex number ${\displaystyle z}$ is expressed in its polar form as ${\displaystyle z=r{e}^{i\theta },}$ its absolute value is ${\displaystyle |z|=r.}$

Since the product of any complex number ${\displaystyle z}$ and its complex conjugate ${\displaystyle \overline{z}=x-iy}$, with the same absolute value, is always the non-negative real number ${\displaystyle \left(x^2+y^2\right)}$, the absolute value of a complex number ${\displaystyle z}$ is the square root of ${\displaystyle z\cdot \overline{z},}$ which is therefore called the absolute square or squared modulus of ${\displaystyle z}$: $${\displaystyle |z|=\sqrt{z\cdot \overline{z}}.}$$ This generalizes the alternative definition for reals: $|x|=\sqrt{x\cdot x}$.

The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity ${\displaystyle |z{|}^2=|z^2|}$ is a special case of multiplicativity that is often useful by itself.

Absolute value function

The graph of the absolute value function for real numbers

Composition of absolute value with a cubic function in different orders

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (−∞, 0] and monotonically increasing on the interval [0, +∞). Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function.

For both real and complex numbers the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself).

Relationship to the sign function

The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:

${\displaystyle |x|=x\mathrm{sgn}(x),}$

or

${\displaystyle |x|\mathrm{sgn}(x)=x,}$

and for x ≠ 0,

${\displaystyle \mathrm{sgn}(x)=\frac{|x|}{x}=\frac{x}{|x|}.}$

Relationship to the max and min functions

Let ${\displaystyle s,t\in \mathbb{R}}$, then the following relationship to the minimum and maximum functions hold:

${\displaystyle |t-s|=-2min(s,t)+s+t}$

and

${\displaystyle |t-s|=2max(s,t)-s-t.}$

The formulas can be derived by considering each case ${\displaystyle s>t}$ and ${\displaystyle t>s}$ separately.

From the last formula one can derive also ${\displaystyle |t|=max(t,-t)}$.

Derivative

The real absolute value function has a derivative for every x ≠ 0, but is not differentiable at x = 0. Its derivative for x ≠ 0 is given by the step function:

${\displaystyle \frac{d\left|x\right|}{dx}=\frac{x}{|x|}=\{\begin{array}{ll}-1& x<0\\ 1& x>0.\end{array}}$

The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist.

The subdifferential of |x| at x = 0 is the interval [−1, 1].

The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.

The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the Dirac delta function.

Antiderivative

The antiderivative (indefinite integral) of the real absolute value function is

${\displaystyle \int \left|x\right|dx=\frac{x\left|x\right|}{2}+C,}$

where C is an arbitrary constant of integration. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not.

Derivatives of compositions

The following two formulae are special cases of the chain rule:

${\displaystyle \frac{d}{dx}f(|x|)=\frac{x}{|x|}(f^{\prime }(|x|))}$

if the absolute value is inside a function, and

${\displaystyle \frac{d}{dx}|f(x)|=\frac{f(x)}{|f(x)|}{f}^{\prime }(x)}$

if another function is inside the absolute value. In the first case, the derivative is always discontinuous at $x=0$ in the first case and where $f(x)=0$ in the second case.

Distance

See also: Metric space

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.

The standard Euclidean distance between two points

${\displaystyle a=(a_1,a_2,\dots ,a_n)}$

and

${\displaystyle b=(b_1,b_2,\dots ,b_n)}$

in Euclidean n-space is defined as:

${\displaystyle \sqrt{\sum _{i=1}^n(a_i-b_i{)}^2}.}$

This can be seen as a generalisation, since for ${\displaystyle a_1}$ and ${\displaystyle b_1}$ real, i.e. in a 1-space, according to the alternative definition of the absolute value,

${\displaystyle |a_1-b_1|=\sqrt{(a_1-b_1{)}^2}=\sqrt{\sum _{i=1}^1(a_i-b_i{)}^2},}$

and for ${\displaystyle a=a_1+i{a}_2}$ and ${\displaystyle b=b_1+i{b}_2}$ complex numbers, i.e. in a 2-space,

${\displaystyle |a-b|}$	${\displaystyle =|(a_1+i{a}_2)-(b_1+i{b}_2)|}$
	${\displaystyle =|(a_1-b_1)+i(a_2-b_2)|}$
	${\displaystyle =\sqrt{(a_1-b_1{)}^2+(a_2-b_2{)}^2}=\sqrt{\sum _{i=1}^2(a_i-b_i{)}^2}.}$

The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively.

The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows:

A real valued function d on a set X × X is called a metric (or a distance function) on X, if it satisfies the following four axioms:

${\displaystyle d(a,b)\ge 0}$	Non-negativity
${\displaystyle d(a,b)=0⟺a=b}$	Identity of indiscernibles
${\displaystyle d(a,b)=d(b,a)}$	Symmetry
${\displaystyle d(a,b)\le d(a,c)+d(c,b)}$	Triangle inequality

Generalizations

Ordered rings

The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if a is an element of an ordered ring R, then the absolute value of a, denoted by |a|, is defined to be:

${\displaystyle |a|=\{\begin{array}{rl}a,& \text{if }a\ge 0\\ -a,& \text{if }a<0.\end{array}}$

where −a is the additive inverse of a, 0 is the additive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring.

Fields

Main article: Absolute value (algebra)

The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued function v on a field F is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms:

${\displaystyle v(a)\ge 0}$	Non-negativity
${\displaystyle v(a)=0⟺a=\mathbf{0}}$	Positive-definiteness
${\displaystyle v(ab)=v(a)v(b)}$	Multiplicativity
${\displaystyle v(a+b)\le v(a)+v(b)}$	Subadditivity or the triangle inequality

Where 0 denotes the additive identity of F. It follows from positive-definiteness and multiplicativity that v(1) = 1, where 1 denotes the multiplicative identity of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(a − b), is a metric and the following are equivalent:

• d satisfies the ultrametric inequality ${\displaystyle d(x,y)\le max(d(x,z),d(y,z))}$ for all x, y, z in F.
• $\left\{v\left(\sum _{k=1}^n\mathbf{1}\right):n\in \mathbb{N}\right\}$ is bounded in R.
• ${\displaystyle v\left(\sum _{k=1}^n\mathbf{1}\right)\le 1}$ for every ${\displaystyle n\in \mathbb{N}}$.
• ${\displaystyle v(a)\le 1\Rightarrow v(1+a)\le 1}$ for all ${\displaystyle a\in F}$.
• ${\displaystyle v(a+b)\le max\{v(a),v(b)\}}$ for all ${\displaystyle a,b\in F}$.

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.

Vector spaces

Main article: Norm (mathematics)

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

A real-valued function on a vector space V over a field F, represented as ‖ · ‖, is called an absolute value, but more usually a norm, if it satisfies the following axioms:

For all a in F, and v, u in V,

${\displaystyle \Vert \mathbf{v}\Vert \ge 0}$	Non-negativity
${\displaystyle \Vert \mathbf{v}\Vert =0⟺\mathbf{v}=0}$	Positive-definiteness
${\displaystyle \Vert a\mathbf{v}\Vert =\left|a\right|\Vert \mathbf{v}\Vert }$	Absolute homogeneity or positive scalability
${\displaystyle \Vert \mathbf{v}+\mathbf{u}\Vert \le \Vert \mathbf{v}\Vert +\Vert \mathbf{u}\Vert }$	Subadditivity or the triangle inequality

The norm of a vector is also called its length or magnitude.

In the case of Euclidean space ${\displaystyle {\mathbb{R}}^n}$, the function defined by

${\displaystyle \Vert (x_1,x_2,\dots ,x_n)\Vert =\sqrt{\sum _{i=1}^n{x}_i^2}}$

is a norm called the Euclidean norm. When the real numbers ${\displaystyle \mathbb{R}}$ are considered as the one-dimensional vector space ${\displaystyle {\mathbb{R}}^1}$, the absolute value is a norm, and is the p-norm (see L^p space) for any p. In fact the absolute value is the "only" norm on ${\displaystyle {\mathbb{R}}^1}$, in the sense that, for every norm ‖ · ‖ on ${\displaystyle {\mathbb{R}}^1}$, ‖x‖ = ‖1‖ ⋅ |x|.

The complex absolute value is a special case of the norm in an inner product space, which is identical to the Euclidean norm when the complex plane is identified as the Euclidean plane ${\displaystyle {\mathbb{R}}^2}$.

Composition algebras

Main article: Composition algebra

Every composition algebra A has an involution x → x* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x.

The real numbers ${\displaystyle \mathbb{R}}$, complex numbers ${\displaystyle \mathbb{C}}$, and quaternions ${\displaystyle \mathbb{H}}$ are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element x has a non-zero norm, then x has a multiplicative inverse given by x*/N(x).

Entropic force

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Entropic_force 

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, the entropic force ${\displaystyle \mathbf{F}}$ associated to a macrostate partition ${\displaystyle \{\mathbf{X}\}}$ is given by

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

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.

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, 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 of molten salt,
• surface tension and,
• elasticity of rubber.

Polymers

Main article: Ideal chain

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

See also: Hydrophobic effect § Cause

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

• donating two hydrogen bonds through the two protons,
• accepting two more hydrogen bonds through the two sp³-hybridized lone 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 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. 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. Structures of water-soluble proteins typically have a core in which hydrophobic side chains 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 colloids, 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. Because of this, entropic forces can be an important driver of self-assembly.

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 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 cytoskeleton. Here, the actin-bundling protein anillin 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, professor of mathematics at Victoria University of Wellington, NZ in "Conservative Entropic Forces" 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

Main article: Entropic 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. The same approach was argued to explain dark matter, dark energy 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.