Post-scarcity

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Post-scarcity

 

Post-scarcity is a theoretical economic situation in which most goods can be produced in great abundance with minimal human labor needed, so that they become available to all very cheaply or even freely.

Post-scarcity does not mean that scarcity has been eliminated for all goods and services but that all people can easily have their basic survival needs met along with some significant proportion of their desires for goods and services. Writers on the topic often emphasize that some commodities will remain scarce in a post-scarcity society.

Models

Speculative technology

Futurists who speak of "post-scarcity" suggest economies based on advances in automated manufacturing technologies, often including the idea of self-replicating machines, the adoption of division of labour which in theory could produce nearly all goods in abundance, given adequate raw materials and energy.

More speculative forms of nanotechnology such as molecular assemblers or nanofactories, which do not currently exist, raise the possibility of devices that can automatically manufacture any specified goods given the correct instructions and the necessary raw materials and energy, and many nanotechnology enthusiasts have suggested it will usher in a post-scarcity world.

In the more near-term future, the increasing automation of physical labor using robots is often discussed as means of creating a post-scarcity economy.

Increasingly versatile forms of rapid prototyping machines, and a hypothetical self-replicating version of such a machine known as a RepRap, have also been predicted to help create the abundance of goods needed for a post-scarcity economy. Advocates of self-replicating machines such as Adrian Bowyer, the creator of the RepRap project, argue that once a self-replicating machine is designed, then since anyone who owns one can make more copies to sell (and would also be free to ask for a lower price than other sellers), market competition will naturally drive the cost of such machines down to the bare minimum needed to make a profit, in this case just above the cost of the physical materials and energy that must be fed into the machine as input, and the same should go for any other goods that the machine can build.

Even with fully automated production, limitations on the number of goods produced would arise from the availability of raw materials and energy, as well as ecological damage associated with manufacturing technologies. Advocates of technological abundance often argue for more extensive use of renewable energy and greater recycling in order to prevent future drops in availability of energy and raw materials, and reduce ecological damage. Solar energy in particular is often emphasized, as the cost of solar panels continues to drop (and could drop far more with automated production by self-replicating machines), and advocates point out the total solar power striking the Earth's surface annually exceeds our civilization's current annual power usage by a factor of thousands.

Advocates also sometimes argue that the energy and raw materials available could be greatly expanded by looking to resources beyond the Earth. For example, asteroid mining is sometimes discussed as a way of greatly reducing scarcity for many useful metals such as nickel. While early asteroid mining might involve crewed missions, advocates hope that eventually humanity could have automated mining done by self-replicating machines. If this were done, then the only capital expenditure would be a single self-replicating unit (whether robotic or nanotechnological), after which the number of units could replicate at no further cost, limited only by the available raw materials needed to build more.

Social

A World Future Society report looked at how historically capitalism takes advantage of scarcity. Increased resource scarcity leads to increase and fluctuation of prices, which drives advances in technology for more efficient use of resources such that costs will be considerably reduced, almost to zero. They thus claim that following an increase in scarcity from now, the world will enter a post-scarcity age between 2050 and 2075.

Murray Bookchin's 1971 essay collection Post-Scarcity Anarchism outlines an economy based on social ecology, libertarian municipalism, and an abundance of fundamental resources, arguing that post-industrial societies have the potential to be developed into post-scarcity societies. Such development would enable "the fulfillment of the social and cultural potentialities latent in a technology of abundance".

Bookchin claims that the expanded production made possible by the technological advances of the twentieth century were in the pursuit of market profit and at the expense of the needs of humans and of ecological sustainability. The accumulation of capital can no longer be considered a prerequisite for liberation, and the notion that obstructions such as the state, social hierarchy, and vanguard political parties are necessary in the struggle for freedom of the working classes can be dispelled as a myth.

Marxism

Karl Marx, in a section of his Grundrisse that came to be known as the "Fragment on Machines", argued that the transition to a post-capitalist society combined with advances in automation would allow for significant reductions in labor needed to produce necessary goods, eventually reaching a point where all people would have significant amounts of leisure time to pursue science, the arts, and creative activities; a state some commentators later labeled as "post-scarcity". Marx argued that capitalism—the dynamic of economic growth based on capital accumulation—depends on exploiting the surplus labor of workers, but a post-capitalist society would allow for:

The free development of individualities, and hence not the reduction of necessary labour time so as to posit surplus labour, but rather the general reduction of the necessary labour of society to a minimum, which then corresponds to the artistic, scientific etc. development of the individuals in the time set free, and with the means created, for all of them.

Marx's concept of a post-capitalist communist society involves the free distribution of goods made possible by the abundance provided by automation. The fully developed communist economic system is postulated to develop from a preceding socialist system. Marx held the view that socialism—a system based on social ownership of the means of production—would enable progress toward the development of fully developed communism by further advancing productive technology. Under socialism, with its increasing levels of automation, an increasing proportion of goods would be distributed freely.

Marx did not believe in the elimination of most physical labor through technological advancements alone in a capitalist society, because he believed capitalism contained within it certain tendencies which countered increasing automation and prevented it from developing beyond a limited point, so that manual industrial labor could not be eliminated until the overthrow of capitalism. Some commentators on Marx have argued that at the time he wrote the Grundrisse, he thought that the collapse of capitalism due to advancing automation was inevitable despite these counter-tendencies, but that by the time of his major work Capital: Critique of Political Economy he had abandoned this view, and came to believe that capitalism could continually renew itself unless overthrown.

Fiction

Literature

• The novella The Midas Plague by Frederik Pohl describes a world of cheap energy, in which robots are overproducing the commodities enjoyed by humankind. The lower-class "poor" must spend their lives in frantic consumption, trying to keep up with the robots' extravagant production, while the upper-class "rich" can live lives of simplicity.
• The Mars trilogy by Kim Stanley Robinson charts the terraforming of Mars as a human colony and the establishment of a post-scarcity society.
• The Culture novels by Iain M. Banks are centered on a post-scarcity economy where technology is advanced to such a degree that all production is automated, and there is no use for money or property (aside from personal possessions with sentimental value). People in the Culture are free to pursue their own interests in an open and socially-permissive society.
  • The society depicted in the Culture novels has been described by some commentators as "communist-bloc" or "anarcho-communist". Banks' close friend and fellow science fiction writer Ken MacLeod has said that The Culture can be seen as a realization of Marx's communism, but adds that "however friendly he was to the radical left, Iain had little interest in relating the long-range possibility of utopia to radical politics in the here and now. As he saw it, what mattered was to keep the utopian possibility open by continuing technological progress, especially space development, and in the meantime to support whatever policies and politics in the real world were rational and humane."
• The Rapture of the Nerds by Cory Doctorow and Charles Stross takes place in a post-scarcity society and involves "disruptive" technology. The title is a derogatory term for the technological singularity coined by SF author Ken MacLeod.
• Con Blomberg's 1959 short story Sales Talk depicts a post-scarcity society in which society incentivizes consumption to reduce the burden of overproduction. To further reduce production, virtual reality is used to fulfill peoples' needs to create.
• Cory Doctorow's novel Walkaway presents a modern take on the idea of post-scarcity. With the advent of 3D printing – and especially the ability to use these to fabricate even better fabricators – and with machines that can search for and reprocess waste or discarded materials, the protagonists no longer have need of regular society for the basic essentials of life, such as food, clothing and shelter.

Television and film

• The 24th-century human society depicted in the television series Star Trek: The Next Generation and Star Trek: Deep Space Nine is a post-scarcity society brought about by the invention of the "replicator", a machine that converts energy to matter instantaneously. In the film Star Trek: First Contact, Captain Jean-Luc Picard asserts: "The acquisition of wealth is no longer the driving force of our lives. We work to better ourselves and the rest of humanity." In this universe (at least the United Federation of Planets), money had been rendered obsolete on Earth by the 22nd century (although it still existed with reference to other species in the Star Trek universe, most notably the Ferengi).

Helmholtz equation

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Helmholtz_equation

In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation:

$${\displaystyle {\mathrm{\nabla }}^{2}f=-k^{2}f,}$$

where ∇² is the Laplace operator, k² is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.

In optics, the Helmholtz equation is the wave equation for the electric field.

The equation is named after Hermann von Helmholtz, who studied it in 1860.

Motivation and uses

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.

For example, consider the wave equation

$${\displaystyle \left({\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)u(\mathbf{r},t)=0.}$$

Separation of variables begins by assuming that the wave function u(r, t) is in fact separable:

$${\displaystyle u(\mathbf{r},t)=A(\mathbf{r})T(t).}$$

Substituting this form into the wave equation and then simplifying, we obtain the following equation:

$${\displaystyle \frac{{\mathrm{\nabla }}^{2}A}{A}=\frac{1}{c^{2}T}\frac{{\mathrm{d}}^{2}T}{\mathrm{d}{t}^2}.}$$

Notice that the expression on the left side depends only on r, whereas the right expression depends only on t. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to the same constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for A(r), the other for T(t):

$${\displaystyle \frac{{\mathrm{\nabla }}^{2}A}{A}=-k^2}$$

$${\displaystyle \frac{1}{c^{2}T}\frac{{\mathrm{d}}^{2}T}{\mathrm{d}{t}^2}=-k^2,}$$

where we have chosen, without loss of generality, the expression −k² for the value of the constant. (It is equally valid to use any constant k as the separation constant; −k² is chosen only for convenience in the resulting solutions.)

Rearranging the first equation, we obtain the Helmholtz equation:

$${\displaystyle {\mathrm{\nabla }}^{2}A+k^{2}A=({\mathrm{\nabla }}^2+k^2)A=0.}$$

Likewise, after making the substitution ω = kc, where k is the wave number, and ω is the angular frequency (assuming a monochromatic field), the second equation becomes

$${\displaystyle \frac{{\mathrm{d}}^{2}T}{\mathrm{d}{t}^2}+{\omega }^{2}T=\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}+{\omega }^2\right)T=0.}$$

We now have Helmholtz's equation for the spatial variable r and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, whose exact form is determined by initial conditions, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.

Because of its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.

Solving the Helmholtz equation using separation of variables

The solution to the spatial Helmholtz equation:

$${\displaystyle {\mathrm{\nabla }}^{2}A=-k^{2}A}$$

can be obtained for simple geometries using separation of variables.

Vibrating membrane

The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred Clebsch in 1862. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieu's differential equation.

If the edges of a shape are straight line segments, then a solution is integrable or knowable in closed-form only if it is expressible as a finite linear combination of plane waves that satisfy the boundary conditions (zero at the boundary, i.e., membrane clamped).

If the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form

$${\displaystyle A_{rr}+\frac{1}{r}{A}_r+\frac{1}{r^2}{A}_{\theta \theta }+k^{2}A=0.}$$

We may impose the boundary condition that A vanishes if r = a; thus

$${\displaystyle A(a,\theta )=0.}$$

the method of separation of variables leads to trial solutions of the form

$${\displaystyle A(r,\theta )=R(r)\mathrm{\Theta }(\theta ),}$$

where Θ must be periodic of period 2π. This leads to

$${\displaystyle {\mathrm{\Theta }}^{″}+n^2\mathrm{\Theta }=0,}$$

$${\displaystyle r^2{R}^{″}+r{R}^{\prime }+r^2{k}^{2}R-n^{2}R=0.}$$

It follows from the periodicity condition that

$${\displaystyle \mathrm{\Theta }=\alpha \cos n\theta +\beta \sin n\theta ,}$$

and that n must be an integer. The radial component R has the form

$${\displaystyle R(r)=\gamma J_n(\rho ),}$$

where the Bessel function J_n(ρ) satisfies Bessel's equation

$${\displaystyle {\rho }^2{J}_n^{″}+\rho J_n^{\prime }+({\rho }^2-n^2)J_n=0,}$$

and ρ = kr. The radial function J_n has infinitely many roots for each value of n, denoted by ρ_m,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by

$${\displaystyle k_{m,n}=\frac{1}{a}{\rho }_{m,n}.}$$

The general solution A then takes the form of a generalized Fourier series of terms involving products of J_n(k_m,nr) and the sine (or cosine) of nθ. These solutions are the modes of vibration of a circular drumhead.

Three-dimensional solutions

In spherical coordinates, the solution is:

$${\displaystyle A(r,\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }\left(a_{\ell m}{j}_{\ell }(kr)+b_{\ell m}{y}_{\ell }(kr)\right)Y_{\ell }^m(\theta ,\phi ).}$$

This solution arises from the spatial solution of the wave equation and diffusion equation. Here j_ℓ(kr) and y_ℓ(kr) are the spherical Bessel functions, and Y^m
_ℓ(θ, φ) are the spherical harmonics (Abramowitz and Stegun, 1964). Note that these forms are general solutions, and require boundary conditions to be specified to be used in any specific case. For infinite exterior domains, a radiation condition may also be required (Sommerfeld, 1949).

Writing r₀ = (x, y, z) function A(r₀) has asymptotics

$${\displaystyle A(r_0)=\frac{e^{ik{r}_0}}{r_0}f\left(\frac{{\mathbf{r}}_0}{r_0},k,u_0\right)+o\left(\frac{1}{r_0}\right)\text{ as }{r}_0\to \mathrm{\infty }}$$

where function f is called scattering amplitude and u₀(r₀) is the value of A at each boundary point r₀.

Three-dimensional solutions given the function on a 2-dimensional plane

Given a 2-dimensional plane where A is known, the solution to the Helmholtz equation is given by:^[3]

$${\displaystyle A(x,y,z)=-\frac{1}{2\pi }{\iint }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{A}^{\prime }(x^{\prime },y^{\prime })\frac{e^{ikr}}{r}\frac{z}{r}\left(ik-\frac{1}{r}\right)d{x}^{\prime }d{y}^{\prime },}$$

where

• ${\displaystyle A^{\prime }(x^{\prime },y^{\prime })}$ is the solution at the 2-dimensional plane,
• ${\displaystyle r=\sqrt{(x-x^{\prime }{)}^2+(y-y^{\prime }{)}^2+z^2},}$

As z approaches zero, all contributions from the integral vanish except for r=0. Thus ${\displaystyle A(x,y,0)=A^{\prime }(x,y)}$ up to a numerical factor, which can be verified to be 1 by transforming the integral to polar coordinates ${\displaystyle (\rho ,\theta )}$.

This solution is important in diffraction theory, e.g. in deriving Fresnel diffraction.

Paraxial approximation

Further information: Slowly varying envelope approximation

In the paraxial approximation of the Helmholtz equation, the complex amplitude A is expressed as

$${\displaystyle A(\mathbf{r})=u(\mathbf{r})e^{ikz}}$$

where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, u approximately solves

$${\displaystyle {\mathrm{\nabla }}_{\perp }^{2}u+2ik\frac{\mathrm{\partial }u}{\mathrm{\partial }z}=0,}$$

where ${\mathrm{\nabla }}_{\perp }^2\stackrel{\text{ def }}{=}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}$ is the transverse part of the Laplacian.

This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves (light) in the form of either paraboloidal waves or Gaussian beams. Most lasers emit beams that take this form.

The assumption under which the paraxial approximation is valid is that the z derivative of the amplitude function u is a slowly varying function of z:

$${\displaystyle \left|\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right|\ll \left|k\frac{\mathrm{\partial }u}{\mathrm{\partial }z}\right|.}$$

This condition is equivalent to saying that the angle θ between the wave vector k and the optical axis z is small: θ ≪ 1.

The paraxial form of the Helmholtz equation is found by substituting the above-stated expression for the complex amplitude into the general form of the Helmholtz equation as follows:

$${\displaystyle {\mathrm{\nabla }}^2(u\left(x,y,z\right)e^{ikz})+k^{2}u\left(x,y,z\right)e^{ikz}=0.}$$

Expansion and cancellation yields the following:

$${\displaystyle \left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\right)u(x,y,z)e^{ikz}+\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}u(x,y,z)\right)e^{ikz}+2\left(\frac{\mathrm{\partial }}{\mathrm{\partial }z}u(x,y,z)\right)ik{e}^{ikz}=0.}$$

Because of the paraxial inequality stated above, the ∂²u/∂z² term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting u(r) = A(r) e^−ikz then gives the paraxial equation for the original complex amplitude A:

$${\displaystyle {\mathrm{\nabla }}_{\perp }^{2}A+2ik\frac{\mathrm{\partial }A}{\mathrm{\partial }z}+2k^{2}A=0.}$$

The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation.

Inhomogeneous Helmholtz equation

Two sources of radiation in the plane, given mathematically by a function f, which is zero in the blue region

 

The real part of the resulting field A, A is the solution to the inhomogeneous Helmholtz equation (∇² + k²) A = −f.

The inhomogeneous Helmholtz equation is the equation

$${\displaystyle {\mathrm{\nabla }}^{2}A(\mathbf{x})+k^{2}A(\mathbf{x})=-f(\mathbf{x})\text{ in }{\mathbb{R}}^n,}$$

where ƒ : Rⁿ → C is a function with compact support, and n = 1, 2, 3. This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the k term) were switched to a minus sign.

In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition

$${\displaystyle \underset{r\to \mathrm{\infty }}{lim}{r}^{\frac{n-1}{2}}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }r}-ik\right)A(\mathbf{x})=0}$$

in ${\displaystyle n}$ spatial dimensions, for all angles (i.e. any value of ${\displaystyle \theta ,\varphi }$). Here ${\displaystyle r=\sqrt{\sum _{i=1}^n{x}_i^2}}$ where ${\displaystyle x_i}$ are the coordinates of the vector ${\displaystyle \mathbf{x}}$.

With this condition, the solution to the inhomogeneous Helmholtz equation is

$${\displaystyle A(\mathbf{x})={\int }_{{\mathbb{R}}^n}G(\mathbf{x},{\mathbf{x}}^{\prime })f({\mathbf{x}}^{\prime })\mathrm{d}{\mathbf{x}}^{\prime }}$$

(notice this integral is actually over a finite region, since f has compact support). Here, G is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with f equaling the Dirac delta function, so G satisfies

$${\displaystyle {\mathrm{\nabla }}^{2}G(\mathbf{x},{\mathbf{x}}^{\prime })+k^{2}G(\mathbf{x},{\mathbf{x}}^{\prime })=-\delta (\mathbf{x},{\mathbf{x}}^{\prime })\in {\mathbb{R}}^n.}$$

The expression for the Green's function depends on the dimension n of the space. One has

$${\displaystyle G(x,x^{\prime })=\frac{i{e}^{ik|x-x^{\prime }|}}{2k}}$$

for n = 1,

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=\frac{i}{4}{H}_0^{(1)}(k|\mathbf{x}-{\mathbf{x}}^{\prime }|)}$$

for n = 2, where H⁽¹⁾
₀ is a Hankel function, and

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=\frac{e^{ik|\mathbf{x}-{\mathbf{x}}^{\prime }|}}{4\pi |\mathbf{x}-{\mathbf{x}}^{\prime }|}}$$

for n = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for |x| → ∞.

Finally, for general n,

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=c_d{k}^p\frac{H_p^{(1)}(k|\mathbf{x}-{\mathbf{x}}^{\prime }|)}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^p}}$$

where ${\displaystyle p=\frac{n-2}{2}}$ and ${\displaystyle c_d=\frac{1}{2i(2\pi {)}^p}}$.