Separation of variables

From Wikipedia, the free encyclopedia

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

 

In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

Solve proportional first order differential equation

Solve linear first order differential equation by separation of variables.

Ordinary differential equations (ODE)

A differential equation for the unknown ${\displaystyle f(x)}$ will be separable if it can be written in the form

${\displaystyle \frac{d}{dx}f(x)=g(x)h(f(x))}$

where ${\displaystyle g}$ and ${\displaystyle h}$ are given functions. This is perhaps more transparent when written using ${\displaystyle y=f(x)}$ as:

${\displaystyle \frac{dy}{dx}=g(x)h(y).}$

So now as long as h(y) ≠ 0, we can rearrange terms to obtain:

${\displaystyle \frac{dy}{h(y)}=g(x)dx,}$

where the two variables x and y have been separated. Note dx (and dy) can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of dx as a differential (infinitesimal) is somewhat advanced.

Alternative notation

Those who dislike Leibniz's notation may prefer to write this as

${\displaystyle \frac{1}{h(y)}\frac{dy}{dx}=g(x),}$

but that fails to make it quite as obvious why this is called "separation of variables". Integrating both sides of the equation with respect to ${\displaystyle x}$, we have

${\displaystyle \int \frac{1}{h(y)}\frac{dy}{dx}dx=\int g(x)dx,}$

(A1)

or equivalently,

${\displaystyle \int \frac{1}{h(y)}dy=\int g(x)dx}$

because of the substitution rule for integrals.

If one can evaluate the two integrals, one can find a solution to the differential equation. Observe that this process effectively allows us to treat the derivative ${\displaystyle \frac{dy}{dx}}$ as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below.

(Note that we do not need to use two constants of integration, in equation (A1) as in

${\displaystyle \int \frac{1}{h(y)}dy+C_1=\int g(x)dx+C_2,}$

because a single constant ${\displaystyle C=C_2-C_1}$ is equivalent.)

Example

Population growth is often modeled by the "logistic" differential equation

${\displaystyle \frac{dP}{dt}=kP\left(1-\frac{P}{K}\right)}$

where ${\displaystyle P}$ is the population with respect to time ${\displaystyle t}$, ${\displaystyle k}$ is the rate of growth, and ${\displaystyle K}$ is the carrying capacity of the environment. Separation of variables now leads to

${\displaystyle \begin{array}{rl}& \int \frac{dP}{P\left(1-P/K\right)}=\int kdt\end{array}}$

which is readily integrated using partial fractions on the left side yielding

${\displaystyle P(t)=\frac{K}{1+A{e}^{-kt}}}$

where A is the constant of integration. We can find ${\displaystyle A}$ in terms of ${\displaystyle P\left(0\right)=P_0}$ at t=0. Noting ${\displaystyle e^0=1}$ we get

${\displaystyle A=\frac{K-P_0}{P_0}.}$

Generalization of separable ODEs to the nth order

Much like one can speak of a separable first-order ODE, one can speak of a separable second-order, third-order or nth-order ODE. Consider the separable first-order ODE:

${\displaystyle \frac{dy}{dx}=f(y)g(x)}$

The derivative can alternatively be written the following way to underscore that it is an operator working on the unknown function, y:

${\displaystyle \frac{dy}{dx}=\frac{d}{dx}(y)}$

Thus, when one separates variables for first-order equations, one in fact moves the dx denominator of the operator to the side with the x variable, and the d(y) is left on the side with the y variable. The second-derivative operator, by analogy, breaks down as follows:

${\displaystyle \frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{d}{dx}(y)\right)}$

The third-, fourth- and nth-derivative operators break down in the same way. Thus, much like a first-order separable ODE is reducible to the form

${\displaystyle \frac{dy}{dx}=f(y)g(x)}$

a separable second-order ODE is reducible to the form

${\displaystyle \frac{d^{2}y}{d{x}^2}=f\left(y^{\prime }\right)g(x)}$

and an nth-order separable ODE is reducible to

${\displaystyle \frac{d^{n}y}{d{x}^n}=f\left(y^{(n-1)}\right)g(x)}$

Example

Consider the simple nonlinear second-order differential equation:

$${\displaystyle y^{″}=(y^{\prime }{)}^2.}$$

This equation is an equation only of y'' and y', meaning it is reducible to the general form described above and is, therefore, separable. Since it is a second-order separable equation, collect all x variables on one side and all y' variables on the other to get:

$${\displaystyle \frac{d(y^{\prime })}{(y^{\prime }{)}^2}=dx.}$$

Now, integrate the right side with respect to x and the left with respect to y':

$${\displaystyle \int \frac{d(y^{\prime })}{(y^{\prime }{)}^2}=\int dx.}$$

This gives

$${\displaystyle -\frac{1}{y^{\prime }}=x+C_1,}$$

which simplifies to:

$${\displaystyle y^{\prime }=-\frac{1}{x+C_1}.}$$

This is now a simple integral problem that gives the final answer:

$${\displaystyle y=C_2-\ln |x+C_1|.}$$

Partial differential equations

See also: Separable partial differential equation

The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.

The analytical method of separation of variables for solving partial differential equations has also been generalized into a computational method of decomposition in invariant structures that can be used to solve systems of partial differential equations.

Example: homogeneous case

Consider the one-dimensional heat equation. The equation is

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\alpha \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}=0}$

(1)

The variable u denotes temperature. The boundary condition is homogeneous, that is

${\displaystyle u{|}_{x=0}=u{|}_{x=L}=0}$

(2)

Let us attempt to find a solution which is not identically zero satisfying the boundary conditions but with the following property: u is a product in which the dependence of u on x, t is separated, that is:

${\displaystyle u(x,t)=X(x)T(t).}$

(3)

Substituting u back into equation (1) and using the product rule,

${\displaystyle \frac{T^{\prime }(t)}{\alpha T(t)}=\frac{X^{″}(x)}{X(x)}.}$

(4)

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −λ. Thus:

${\displaystyle T^{\prime }(t)=-\lambda \alpha T(t),}$

(5)

and

${\displaystyle X^{″}(x)=-\lambda X(x).}$

(6)

−λ here is the eigenvalue for both differential operators, and T(t) and X(x) are corresponding eigenfunctions.

We will now show that solutions for X(x) for values of λ ≤ 0 cannot occur:

Suppose that λ < 0. Then there exist real numbers B, C such that

${\displaystyle X(x)=B{e}^{\sqrt{-\lambda }x}+C{e}^{-\sqrt{-\lambda }x}.}$

From (2) we get

${\displaystyle X(0)=0=X(L),}$

(7)

and therefore B = 0 = C which implies u is identically 0.

Suppose that λ = 0. Then there exist real numbers B, C such that

${\displaystyle X(x)=Bx+C.}$

From (7) we conclude in the same manner as in 1 that u is identically 0.

Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that

${\displaystyle T(t)=A{e}^{-\lambda \alpha t},}$

and

${\displaystyle X(x)=B\sin (\sqrt{\lambda }x)+C\cos (\sqrt{\lambda }x).}$

From (7) we get C = 0 and that for some positive integer n,

${\displaystyle \sqrt{\lambda }=n\frac{\pi }{L}.}$

This solves the heat equation in the special case that the dependence of u has the special form of (3).

In general, the sum of solutions to (1) which satisfy the boundary conditions (2) also satisfies (1) and (3). Hence a complete solution can be given as

${\displaystyle u(x,t)=\sum _{n=1}^{\mathrm{\infty }}{D}_n\sin \frac{n\pi x}{L}\exp \left(-\frac{n^2{\pi }^2\alpha t}{L^2}\right),}$

where D_n are coefficients determined by initial condition.

Given the initial condition

${\displaystyle u{|}_{t=0}=f(x),}$

we can get

${\displaystyle f(x)=\sum _{n=1}^{\mathrm{\infty }}{D}_n\sin \frac{n\pi x}{L}.}$

This is the sine series expansion of f(x) which is amenable to Fourier analysis. Multiplying both sides with $\sin \frac{n\pi x}{L}$ and integrating over [0, L] results in

${\displaystyle D_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \frac{n\pi x}{L}dx.}$

This method requires that the eigenfunctions X, here ${\left\{\sin \frac{n\pi x}{L}\right\}}_{n=1}^{\mathrm{\infty }}$, are orthogonal and complete. In general this is guaranteed by Sturm–Liouville theory.

Example: nonhomogeneous case

Suppose the equation is nonhomogeneous,

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\alpha \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}=h(x,t)}$

(8)

with the boundary condition the same as (2).

Expand h(x,t), u(x,t) and f(x) into

${\displaystyle h(x,t)=\sum _{n=1}^{\mathrm{\infty }}{h}_n(t)\sin \frac{n\pi x}{L},}$

(9)

${\displaystyle u(x,t)=\sum _{n=1}^{\mathrm{\infty }}{u}_n(t)\sin \frac{n\pi x}{L},}$

(10)

${\displaystyle f(x)=\sum _{n=1}^{\mathrm{\infty }}{b}_n\sin \frac{n\pi x}{L},}$

(11)

where h_n(t) and b_n can be calculated by integration, while u_n(t) is to be determined.

Substitute (9) and (10) back to (8) and considering the orthogonality of sine functions we get

${\displaystyle u_n^{\prime }(t)+\alpha \frac{n^2{\pi }^2}{L^2}{u}_n(t)=h_n(t),}$

which are a sequence of linear differential equations that can be readily solved with, for instance, Laplace transform, or Integrating factor. Finally, we can get

${\displaystyle u_n(t)=e^{-\alpha \frac{n^2{\pi }^2}{L^2}t}\left(b_n+{\int }_0^t{h}_n(s)e^{\alpha \frac{n^2{\pi }^2}{L^2}s}ds\right).}$

If the boundary condition is nonhomogeneous, then the expansion of (9) and (10) is no longer valid. One has to find a function v that satisfies the boundary condition only, and subtract it from u. The function u-v then satisfies homogeneous boundary condition, and can be solved with the above method.

Example: mixed derivatives

For some equations involving mixed derivatives, the equation does not separate as easily as the heat equation did in the first example above, but nonetheless separation of variables may still be applied. Consider the two-dimensional biharmonic equation

${\displaystyle \frac{{\mathrm{\partial }}^{4}u}{\mathrm{\partial }{x}^4}+2\frac{{\mathrm{\partial }}^{4}u}{\mathrm{\partial }{x}^2\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{4}u}{\mathrm{\partial }{y}^4}=0.}$

Proceeding in the usual manner, we look for solutions of the form

${\displaystyle u(x,y)=X(x)Y(y)}$

and we obtain the equation

${\displaystyle \frac{X^{(4)}(x)}{X(x)}+2\frac{X^{″}(x)}{X(x)}\frac{Y^{″}(y)}{Y(y)}+\frac{Y^{(4)}(y)}{Y(y)}=0.}$

Writing this equation in the form

${\displaystyle E(x)+F(x)G(y)+H(y)=0,}$

Taking the derivative of this expression with respect to ${\displaystyle x}$ gives ${\displaystyle E^{\prime }(x)+F^{\prime }(x)G(y)=0}$ which means ${\displaystyle G(y)=const.}$ or ${\displaystyle F^{\prime }(x)=0}$ and likewise, taking derivative with respect to ${\displaystyle y}$ leads to ${\displaystyle F(x)G^{\prime }(y)+H^{\prime }(y)=0}$ and thus ${\displaystyle F(x)=const.}$ or ${\displaystyle G^{\prime }(y)=0}$, hence either F(x) or G(y) must be a constant, say −λ. This further implies that either ${\displaystyle -E(x)=F(x)G(y)+H(y)}$ or ${\displaystyle -H(y)=E(x)+F(x)G(y)}$ are constant. Returning to the equation for X and Y, we have two cases

${\displaystyle \begin{array}{rl}{X}^{″}(x)& =-{\lambda }_{1}X(x)\\ X^{(4)}(x)& ={\mu }_{1}X(x)\\ Y^{(4)}(y)-2{\lambda }_1{Y}^{″}(y)& =-{\mu }_{1}Y(y)\end{array}}$

and

${\displaystyle \begin{array}{rl}{Y}^{″}(y)& =-{\lambda }_{2}Y(y)\\ Y^{(4)}(y)& ={\mu }_{2}Y(y)\\ X^{(4)}(x)-2{\lambda }_2{X}^{″}(x)& =-{\mu }_{2}X(x)\end{array}}$

which can each be solved by considering the separate cases for ${\displaystyle {\lambda }_i<0,{\lambda }_i=0,{\lambda }_i>0}$ and noting that ${\displaystyle {\mu }_i={\lambda }_i^2}$.

Curvilinear coordinates

In orthogonal curvilinear coordinates, separation of variables can still be used, but in some details different from that in Cartesian coordinates. For instance, regularity or periodic condition may determine the eigenvalues in place of boundary conditions. See spherical harmonics for example.

Applicability

Partial differential equations

For many PDEs, such as the wave equation, Helmholtz equation and Schrodinger equation, the applicability of separation of variables is a result of the spectral theorem. In some cases, separation of variables may not be possible. Separation of variables may be possible in some coordinate systems but not others, and which coordinate systems allow for separation depends on the symmetry properties of the equation. Below is an outline of an argument demonstrating the applicability of the method to certain linear equations, although the precise method may differ in individual cases (for instance in the biharmonic equation above).

Consider an initial boundary value problem for a function ${\displaystyle u(x,t)}$ on ${\displaystyle D=\{(x,t):x\in [0,l],t\ge 0\}}$ in two variables:

${\displaystyle (Tu)(x,t)=(Su)(x,t)}$

where ${\displaystyle T}$ is a differential operator with respect to ${\displaystyle x}$ and ${\displaystyle S}$ is a differential operator with respect to ${\displaystyle t}$ with boundary data:

${\displaystyle (Tu)(0,t)=(Tu)(l,t)=0}$ for ${\displaystyle t\ge 0}$

${\displaystyle (Su)(x,0)=h(x)}$ for ${\displaystyle 0\le x\le l}$

where ${\displaystyle h}$ is a known function.

We look for solutions of the form ${\displaystyle u(x,t)=f(x)g(t)}$. Dividing the PDE through by ${\displaystyle f(x)g(t)}$ gives

${\displaystyle \frac{Tf}{f}=\frac{Sg}{g}}$

The right hand side depends only on ${\displaystyle x}$ and the left hand side only on ${\displaystyle t}$ so both must be equal to a constant ${\displaystyle K}$, which gives two ordinary differential equations

${\displaystyle Tf=Kf,Sg=Kg}$

which we can recognize as eigenvalue problems for the operators for ${\displaystyle T}$ and ${\displaystyle S}$. If ${\displaystyle T}$ is a compact, self-adjoint operator on the space ${\displaystyle L^2[0,l]}$ along with the relevant boundary conditions, then by the Spectral theorem there exists a basis for ${\displaystyle L^2[0,l]}$ consisting of eigenfunctions for ${\displaystyle T}$. Let the spectrum of ${\displaystyle T}$ be ${\displaystyle E}$ and let ${\displaystyle f_{\lambda }}$ be an eigenfunction with eigenvalue ${\displaystyle \lambda \in E}$. Then for any function which at each time ${\displaystyle t}$ is square-integrable with respect to ${\displaystyle x}$, we can write this function as a linear combination of the ${\displaystyle f_{\lambda }}$. In particular, we know the solution ${\displaystyle u}$ can be written as

${\displaystyle u(x,t)=\sum _{\lambda \in E}{c}_{\lambda }(t)f_{\lambda }(x)}$

For some functions ${\displaystyle c_{\lambda }(t)}$. In the separation of variables, these functions are given by solutions to ${\displaystyle Sg=Kg}$

Hence, the spectral theorem ensures that the separation of variables will (when it is possible) find all the solutions.

For many differential operators, such as ${\displaystyle \frac{d^2}{d{x}^2}}$, we can show that they are self-adjoint by integration by parts. While these operators may not be compact, their inverses (when they exist) may be, as in the case of the wave equation, and these inverses have the same eigenfunctions and eigenvalues as the original operator (with the possible exception of zero).

Matrices

The matrix form of the separation of variables is the Kronecker sum.

As an example we consider the 2D discrete Laplacian on a regular grid:

${\displaystyle L={\mathbf{D}}_{\mathbf{x}\mathbf{x}}\oplus {\mathbf{D}}_{\mathbf{y}\mathbf{y}}={\mathbf{D}}_{\mathbf{x}\mathbf{x}}\otimes \mathbf{I}+\mathbf{I}\otimes {\mathbf{D}}_{\mathbf{y}\mathbf{y}},}$

where ${\displaystyle {\mathbf{D}}_{\mathbf{x}\mathbf{x}}}$ and ${\displaystyle {\mathbf{D}}_{\mathbf{y}\mathbf{y}}}$ are 1D discrete Laplacians in the x- and y-directions, correspondingly, and ${\displaystyle \mathbf{I}}$ are the identities of appropriate sizes. See the main article Kronecker sum of discrete Laplacians for details.

Software

Some mathematical programs are able to do separation of variables: Xcas^[6] among others.

Sacrifice zone

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

A sacrifice zone where Iron hydroxide precipitate from coal mining has damaged a stream and surrounding area

A sacrifice zone or sacrifice area (often termed a national sacrifice zone or national sacrifice area) is a geographic area that has been permanently changed by heavy environmental alterations (often to a negative degree) or economic disinvestment, often through locally unwanted land use (LULU). Commentators including Chris Hedges, Joe Sacco, and Steve Lerner have argued that corporate business practices contribute to producing sacrifice zones. A 2022 report by the United Nations highlighted that millions of people globally are in pollution sacrifice zones, particularly in zones used for heavy industry and mining.

Definition

A sacrifice zone or sacrifice area is a geographic area that has been permanently impaired by environmental damage or economic disinvestment.

Another definition states that sacrifice zones are places damaged through locally unwanted land use causing "chemical pollution where residents live immediately adjacent to heavily polluted industries or military bases."

Origin of the concept

Juskus (2023) refers that the concept of Sacrifice Zone has its origins in the field livestock management, being used to refer to the spaces where farmers concentrated cattle waste in order to protect the remaining pasture land. However, the concept would be appropriated by the American Indian Movement and some environmentalist struggles transforming it from a technical term used for land and animal management to a way of conceiving geographical spaces in which the destruction of natural resources is a problem.

According to Helen Huntington Smith, the term was first used in the U.S. discussing the long-term effects of strip-mining coal in the American West in the 1970s. The National Academy of Sciences/National Academy of Engineering Study Committee on the Potential for Rehabilitating Lands Surface Mined for Coal in the Western United States produced a 1973 report that introduced the term, finding:

In each zone the probability of rehabilitating an area depends upon the land use objectives, the characteristics of the site, the technology available, and the skill with which this technology is applied. At the extremes, if surface mined lands are declared national sacrifice areas, all ecological zones have a high probability of being successfully rehabilitated. If, however, complete restoration is the objective, rehabilitation in each zone has no probability of success.

Similarly in 1975, Genevieve Atwood wrote in Scientific American:

Surface mining without reclamation removes the land forever from productive use; such land can best be classified as a national sacrifice area. With successful reclamation, however, surface mining can become just one of a series of land uses that merely interrupt a current use and then return the land to an equivalent potential productivity or an even higher one.

Huntington Smith wrote in 1975, "The Panel that issued the cautious and scholarly National Academy of Sciences report unwitting touched off a verbal bombshell" with the phrase National Sacrifice Area; "The words exploded in the Western press overnight. Seized upon by a people who felt themselves being served up as 'national sacrifices,' they became a watchword and a rallying cry."^[ The term sparked public debate, including among environmentalists and politicians such as future Colorado governor Richard Lamm

The term continued to be used in the context of strip mining until at least 1999: "West Virginia has become an environmental sacrifice zone".

Cases

Argentina

Further information: Flammable: Environmental Suffering in an Argentine Shantytown

Villa Inflamable neighborhood is located in the city of Dock Sud and is part of the Greater Buenos Aires Metropolitan Area.  The community is situated at the center of a petrochemical development area, where 44 hydrocarbon companies are currently operating. These same companies are mainly responsible for turning the Riachuelo-Matanza basin into one of the most polluted bodies of water in the world.

Reports from Argentinian and foreign public agencies have confirmed the presence of lead, chromium, benzene and other hazardous chemicals in the water supplies of the neighborhood, in amounts far in excess of what is allowed by international regulations. Journalistic and academic research has collected multiple testimonies of serious health diagnoses commonly associated with the presence of these contaminants.   Likewise the book Flammable: Environmental Suffering in an Argentine Shantytown explores the effects of toxicity in the daily lives of the residents of the Inflammable neighborhood, referring to multiple diagnoses of lead poisoning among the inhabitants of Inflammable, especially among children.

Chile

Further information: Pollution in Quintero and Puchuncaví

Reportedly, in 2011 Terram introduced the term sacrifice zone to the Chilean political discourse.

The Chilean port of Quintero and adjacent Puchuncaví have been pointed out as a sacrifice zone. The zone hosts the coal-fired Ventanas Power Plant, an oil refinery, a cement storage, Fundición Ventanas, a copper foundry and refinery, a lubricant factory and a chemical terminal. In total 15 polluting companies operate in the area. In 2011, Escuela La Greda located in Puchuncaví, was engulfed in a chemical cloud from the Ventanas Industrial Complex. The sulfur cloud poisoned an estimated 33 children and 9 teachers, resulting in the relocation of the school. The old location of the school is now abandoned. In August and September 2018 there was a public health crisis in Quintero and Puchuncaví, where over 300 people experienced illness from toxic substances in the air, coming from the polluting industries.

Mexico

The Endhó Dam, often referred to as the "largest septic tank in Latin America" is a heavily polluted body of water that was built in the 1950s to supply irrigation water to the Mezquital Valley region of the State of Hidalgo and today receives about 70% of Mexico City's sewage effluent. The river that feeds the dam is also a major repository for industrial waste from an oil refinery, two large cement factories, and several industrial parks in the region.  These sources of pollution have spread to nearby springs affecting people, animals and crops.

Contamination at the Endhó Dam. The banks of the dam show a turquoise green color due to the presence of heavy metals in the water in addition to the accumulation of multiple solid wastes that are carried by the drainage system of the Mexico City metropolitan area.

Journalist Carlos Carabaña indicates that since 2007, the National Water Commission has issued reports to municipal and state authorities repeatedly informing them of the presence of high levels of heavy metals in the nearby wells, urging the authorities to take action because of the potential health risks posed by the dam. Other effects related to contamination from the dam include damage to crops in the communities neighboring the dam, poisoning of livestock, and stigmatization of agricultural products from the Mezquital Valley region.

United States

See also: Superfund

The US EPA affirmed in a 2004 report in response to the Office of Inspector General, that "the solution to unequal protection lies in the realm of environmental justice for all Americans. No community, rich or poor, black or white, should be allowed to become a 'sacrifice zone'."

Commentators including Chris Hedges, Joe Sacco, Robert Bullard and Stephen Lerner have argued that corporate business practices contribute to producing sacrifice zones and that these zones most commonly exist in low-income and minority, usually African-American communities. Sacrifice zones are a central topic for the graphic novel Days of Destruction, Days of Revolt, written by Hedges and illustrated by Sacco.

In 2012, Hedges stated that examples of sacrifice zones included Pine Ridge, South Dakota and Camden, New Jersey In 2017 a West Calumet public housing project in East Chicago, Indiana built at the former site of a lead smelter needed to be demolished and soil replaced to bring the area up to residential standards, displacing 1000 residents. In 2014, Naomi Klein wrote that "running an economy on energy sources that release poisons as an unavoidable part of their extraction and refining has always required sacrifice zones."

Venezuela

Pollution at Maracaibo Lake

Lake Maracaibo in the state of Zulia is one of the most important bodies of water in the western region of Venezuela.  This lake was also the site of one of the worst environmental catastrophes in Venezuela's history: the Barroso II blowout in 1922; an oil well that began spewing huge quantities of oil for 9 days, spilling around 900,000 barrels in the area. This oil disaster, paradoxically, became a milestone for the abundance of the oil industry in the country.

Erick Camargo indicates that oil spills generated by the lack of maintenance of the complex network of oil infrastructure continue to be a constant and are the main cause of contamination in the lake.  However, he also indicates that the use of agrochemicals on nearby crops and the discharge of sewage worsen the situation.

A 2022 scientific paper reveals the presence of multiple toxic elements in surface sediments in different parts of the lake.  This constitutes a high risk for the flora and fauna of the region, as well as for the health of the human communities living in the areas where the samples were taken. Another study in 2007 revealed the presence of toxic metals in part of the subway aquifers connected to the lake basin; the samples taken had values well above the limits allowed for drinking water according to national and international regulations.

Space industry

Point Nemo is also known as “the oceanic pole of inaccessibility” for being the point in any ocean farthest from land. It serves as a "spacecraft cemetery" for space infrastructure and vessels.

The human-environment interactions that lie at the heart of environmental justice, including sacrifice zones, have been proposed to also include the environmental sacrifice of regions beyond Earth. Klinger states that "the environmental geopolitics of Earth and outer space are inextricably linked by the spatial politics of privilege and sacrifice - among people, places and institutions". Dunnett has called outer space the 'ultimate sacrifice zone' that exemplifies a colonially framed pursuit of infinite opportunities for accumulation, exploitation, and pollution. This manifests in both terrestrial and space-based sacrifice zones related to launch infrastructure, waste, and orbital debris.

Point Nemo is an oceanic point of inaccessibility located inside the South Pacific Gyre. It is selected as the most remote location in the world and serves as a "spacecraft cemetery" for space infrastructure and vessels. Since 1971, 273 spacecraft and satellites have been directed to Point Nemo; this number includes the Mir Space Station (142 tonnes) and will include the International Space Station (240 tonnes).