Brownian motion

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2-dimensional random walk of a silver adatom on an Ag(111) surface

Simulation of the Brownian motion of a large particle, analogous to a dust particle, that collides with a large set of smaller particles, analogous to molecules of a gas, which move with different velocities in different random directions.

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas).

This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem).

This motion is named after the Scottish botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant Clarkia pulchella immersed in water. In 1900, the French mathematician Louis Bachelier modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (Théorie de la spéculation), prepared under the supervision of Henri Poincaré. Then, in 1905, theoretical physicist Albert Einstein published a paper where he modeled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions.

The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter".

The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the limit) to Brownian motion (see random walk and Donsker's theorem).

History

Reproduced from the book of Jean Baptiste Perrin, Les Atomes, three tracings of the motion of colloidal particles of radius 0.53 μm, as seen under the microscope, are displayed. Successive positions every 30 seconds are joined by straight line segments (the mesh size is 3.2 μm).

The Roman philosopher-poet Lucretius' scientific poem "On the Nature of Things" (c. 60 BC) has a remarkable description of the motion of dust particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms:

Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible.

Although the mingling, tumbling motion of dust particles is caused largely by air currents, the glittering, jiggling motion of small dust particles is caused chiefly by true Brownian dynamics; Lucretius "perfectly describes and explains the Brownian movement by a wrong example".

While Jan Ingenhousz described the irregular motion of coal dust particles on the surface of alcohol in 1785, the discovery of this phenomenon is often credited to the botanist Robert Brown in 1827. Brown was studying pollen grains of the plant Clarkia pulchella suspended in water under a microscope when he observed minute particles, ejected by the pollen grains, executing a jittery motion. By repeating the experiment with particles of inorganic matter he was able to rule out that the motion was life-related, although its origin was yet to be explained.

The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880. This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. The Brownian model of financial markets is often cited, but Benoit Mandelbrot rejected its applicability to stock price movements in part because these are discontinuous.

Albert Einstein (in one of his 1905 papers) and Marian Smoluchowski (1906) brought the solution of the problem to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules. Their equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in 1908.

The instantaneous velocity of the Brownian motion can be defined as v = Δx/Δt, when Δt << τ, where τ is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air with optical tweezers) was measured successfully. The velocity data verified the Maxwell–Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle.

Statistical mechanics theories

Einstein's theory

There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities. In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular weight in grams, of a gas. In accordance to Avogadro's law, this volume is the same for all ideal gases, which is 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the Avogadro number, and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the molar mass of the gas by the Avogadro constant.

The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a Dirac delta function, indicating that all the particles are located at the origin at time t = 0. As t increases, the distribution flattens (though remains bell-shaped), and ultimately becomes uniform in the limit that time goes to infinity.

The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 10¹⁴ collisions per second.

He regarded the increment of particle positions in time ${\displaystyle \tau }$ in a one-dimensional (x) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable (${\displaystyle q}$) with some probability density function ${\displaystyle \phi (q)}$ (i.e., ${\displaystyle \phi (q)}$ is the probability density for a jump of magnitude ${\displaystyle q}$, i.e., the probability density of the particle incrementing its position from ${\displaystyle x}$ to ${\displaystyle x+q}$ in the time interval ${\displaystyle \tau }$). Further, assuming conservation of particle number, he expanded the number density ${\displaystyle \rho (x,t+\tau )}$ (number of particles per unit volume around ${\displaystyle x}$) at time ${\displaystyle t+\tau }$ in a Taylor series, $${\displaystyle \begin{array}{rl}\rho (x,t+\tau )=& \rho (x,t)+\tau \frac{\mathrm{\partial }\rho (x,t)}{\mathrm{\partial }t}+\cdots \\ =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\rho (x-q,t)\phi (q)dq={\mathbb{E}}_q\left[\rho (x-q,t)\right]\\ =& \rho (x,t){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\phi (q)dq-\frac{\mathrm{\partial }\rho }{\mathrm{\partial }x}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}q\phi (q)dq+\frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{x}^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{q^2}{2}\phi (q)dq+\cdots \\ =& \rho (x,t)\cdot 1-0+\frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{x}^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{q^2}{2}\phi (q)dq+\cdots \end{array}}$$ where the second equality is by definition of ${\displaystyle \phi }$. The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments) vanish because of space symmetry. What is left gives rise to the following relation: $${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}=\frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{x}^2}\cdot {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{q^2}{2\tau }\phi (q)dq+\text{higher-order even moments.}}$$ Where the coefficient after the Laplacian, the second moment of probability of displacement ${\displaystyle q}$, is interpreted as mass diffusivity D: $${\displaystyle D={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{q^2}{2\tau }\phi (q)dq.}$$ Then the density of Brownian particles ρ at point x at time t satisfies the diffusion equation: $${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}=D\cdot \frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{x}^2},}$$

Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution $${\displaystyle \rho (x,t)=\frac{N}{\sqrt{4\pi Dt}}\exp \left(-\frac{x^2}{4Dt}\right).}$$ This expression (which is a normal distribution with the mean ${\displaystyle \mu =0}$ and variance ${\displaystyle {\sigma }^2=2Dt}$ usually called Brownian motion ${\displaystyle B_t}$) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by $${\displaystyle \mathbb{E}\left[x^2\right]=2Dt.}$$ This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root. His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.

The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium.

In his original treatment, Einstein considered an osmotic pressure experiment, but the same conclusion can be reached in other ways.

Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of v = μmg, where m is the mass of the particle, g is the acceleration due to gravity, and μ is the particle's mobility in the fluid. George Stokes had shown that the mobility for a spherical particle with radius r is ${\displaystyle \mu =\frac{1}{6\pi \eta r}}$, where η is the dynamic viscosity of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the barometric distribution $${\displaystyle \rho ={\rho }_o\exp \left(-\frac{mgh}{k_{\text{B}}T}\right),}$$ where ρ − ρ_o is the difference in density of particles separated by a height difference, of ${\displaystyle h=z-z_o}$, k_B is the Boltzmann constant (the ratio of the universal gas constant, R, to the Avogadro constant, N_A), and T is the absolute temperature.

Perrin examined the equilibrium (barometric distribution) of granules (0.6 microns) of gamboge, a viscous substance, under the microscope. The granules move against gravity to regions of lower concentration. The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air.

Dynamic equilibrium is established because the more that particles are pulled down by gravity, the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by Fick's law, $${\displaystyle J=-D\frac{d\rho }{dh},}$$ where J = ρv. Introducing the formula for ρ, we find that $${\displaystyle v=\frac{Dmg}{k_{\text{B}}T}.}$$

In a state of dynamical equilibrium, this speed must also be equal to v = μmg. Both expressions for v are proportional to mg, reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical charged particles of charge q in a uniform electric field of magnitude E, where mg is replaced with the electrostatic force qE. Equating these two expressions yields the Einstein relation for the diffusivity, independent of mg or qE or other such forces: $${\displaystyle \frac{\mathbb{E}\left[x^2\right]}{2t}=D=\mu k_{\text{B}}T=\frac{\mu RT}{N_{\text{A}}}=\frac{RT}{6\pi \eta r{N}_{\text{A}}}.}$$ Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the Boltzmann constant as k_B = R / N_A, and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant R, the temperature T, the viscosity η, and the particle radius r, the Avogadro constant N_A can be determined.

The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by J. J. Thomson in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a concentration gradient given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other".

An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in 1888 in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by Stokes's law. He writes ${\displaystyle k^{\prime }=p_o/k}$ for the diffusion coefficient k′, where ${\displaystyle p_o}$ is the osmotic pressure and k is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the ideal gas law per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's. The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the mean free path.

At first, the predictions of Einstein's formula were seemingly refuted by a series of experiments by Svedberg in 1906 and 1907, which gave displacements of the particles as 4 to 6 times the predicted value, and by Henri in 1908 who found displacements 3 times greater than Einstein's formula predicted. But Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909. The confirmation of Einstein's theory constituted empirical progress for the kinetic theory of heat. In essence, Einstein showed that the motion can be predicted directly from the kinetic model of thermal equilibrium. The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the second law of thermodynamics as being an essentially statistical law.

Smoluchowski model

Smoluchowski's theory of Brownian motion starts from the same premise as that of Einstein and derives the same probability distribution ρ(x, t) for the displacement of a Brownian particle along the x in time t. He therefore gets the same expression for the mean squared displacement: ${\displaystyle \mathbb{E}\left[(\mathrm{\Delta }x{)}^2\right]}$. However, when he relates it to a particle of mass m moving at a velocity u which is the result of a frictional force governed by Stokes's law, he finds $${\displaystyle \mathbb{E}\left[(\mathrm{\Delta }x{)}^2\right]=2Dt=t\frac{32}{81}\frac{m{u}^2}{\pi \mu a}=t\frac{64}{27}\frac{\frac{1}{2}m{u}^2}{3\pi \mu a},}$$ where μ is the viscosity coefficient, and a is the radius of the particle. Associating the kinetic energy ${\displaystyle m{u}^2/2}$ with the thermal energy RT/N, the expression for the mean squared displacement is 64/27 times that found by Einstein. The fraction 27/64 was commented on by Arnold Sommerfeld in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."

Smoluchowski attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal. If the probability of m gains and n − m losses follows a binomial distribution, $${\displaystyle P_{m,n}=(\genfrac{}{}{0ex}{}{n}{m})2^{-n},}$$ with equal a priori probabilities of 1/2, the mean total gain is $${\displaystyle \mathbb{E}\left[2m-n\right]=\sum _{m=\frac{n}{2}}^n(2m-n)P_{m,n}=\frac{nn!}{2^{n+1}{\left[\left(\frac{n}{2}\right)!\right]}^2}.}$$

If n is large enough so that Stirling's approximation can be used in the form $${\displaystyle n!\approx {\left(\frac{n}{e}\right)}^n\sqrt{2\pi n},}$$ then the expected total gain will be $${\displaystyle \mathbb{E}\left[2m-n\right]\approx \sqrt{\frac{n}{2\pi }},}$$ showing that it increases as the square root of the total population.

Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a speed u. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be mu/M. This ratio is of the order of 10⁻⁷ cm/s. But we also have to take into consideration that in a gas there will be more than 10¹⁶ collisions in a second, and even greater in a liquid where we expect that there will be 10²⁰ collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 10⁸ to 10¹⁰ collisions in one second, then velocity of the Brownian particle may be anywhere between 10–1000 cm/s. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.

These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, U, which depends on the collisions that tend to accelerate and decelerate it. The larger U is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of energy applies, the kinetic energy of the Brownian particle, ${\displaystyle M{U}^2/2}$, will be equal, on the average, to the kinetic energy of the surrounding fluid particle, ${\displaystyle m{u}^2/2}$.

In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion. The model assumes collisions with M ≫ m where M is the test particle's mass and m the mass of one of the individual particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision always imparts the same magnitude of ΔV. If N_R is the number of collisions from the right and N_L the number of collisions from the left then after N collisions the particle's velocity will have changed by ΔV(2N_R − N). The multiplicity is then simply given by: $${\displaystyle (\genfrac{}{}{0ex}{}{N}{N_{\text{R}}})=\frac{N!}{N_{\text{R}}!(N-N_{\text{R}})!}}$$ and the total number of possible states is given by 2^N. Therefore, the probability of the particle being hit from the right N_R times is: $${\displaystyle P_N(N_{\text{R}})=\frac{N!}{2^N{N}_{\text{R}}!(N-N_{\text{R}})!}}$$

As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions don't apply. For example, the assumption that on average occurs an equal number of collisions from the right as from the left falls apart once the particle is in motion. Also, there would be a distribution of different possible ΔVs instead of always just one in a realistic situation.

Langevin equation

Main article: Langevin equation

The diffusion equation yields an approximation of the time evolution of the probability density function associated with the position of the particle going under a Brownian movement under the physical definition. The approximation is valid on short timescales. The time evolution of the position of the Brownian particle over all time scales described using the Langevin equation, an equation that involves a random force field representing the effect of the thermal fluctuations of the solvent on the particle. In Langevin dynamics and Brownian dynamics, the Langevin equation is used to efficiently simulate the dynamics of molecular systems that exhibit a strong Brownian component.

Astrophysics: star motion within galaxies

In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. The rms velocity V of the massive object, of mass M, is related to the rms velocity ${\displaystyle v_{\star }}$ of the background stars by $${\displaystyle M{V}^2\approx m{v}_{\star }^2}$$ where ${\displaystyle m\ll M}$ is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both ${\displaystyle v_{\star }}$ and V. The Brownian velocity of Sgr A*, the supermassive black hole at the center of the Milky Way galaxy, is predicted from this formula to be less than 1 km s⁻¹.

Mathematics

Main article: Wiener process

In mathematics, Brownian motion is described by the Wiener process, a continuous-time stochastic process named in honor of Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics.

A single realisation of three-dimensional Brownian motion for times 0 ≤ t ≤ 2

The Wiener process W_t is characterized by four facts:

1. W₀ = 0
2. W_t is almost surely continuous
3. W_t has independent increments
4. ${\displaystyle W_t-W_s\sim \mathcal{N}(0,t-s)}$ (for ${\displaystyle 0\le s\le t}$).

${\displaystyle \mathcal{N}(\mu ,{\sigma }^2)}$ denotes the normal distribution with expected value μ and variance σ². The condition that it has independent increments means that if ${\displaystyle 0\le s_1<t_1\le s_2<t_2}$ then ${\displaystyle W_{t_1}-W_{s_1}}$ and ${\displaystyle W_{t_2}-W_{s_2}}$ are independent random variables. In addition, for some filtration ${\displaystyle {\mathcal{F}}_t}$, ${\displaystyle W_t}$ is ${\displaystyle {\mathcal{F}}_t}$ measurable for all ${\displaystyle t\ge 0}$.

An alternative characterisation of the Wiener process is the so-called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with W₀ = 0 and quadratic variation ${\displaystyle [W_t,W_t]=t}$.

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent ${\displaystyle \mathcal{N}(0,1)}$ random variables. This representation can be obtained using the Kosambi–Karhunen–Loève theorem.

The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant. A d-dimensional Gaussian free field has been described as "a d-dimensional-time analog of Brownian motion."

Statistics

The Brownian motion can be modeled by a random walk.

In the general case, Brownian motion is a Markov process and described by stochastic integral equations.

Lévy characterisation

The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous Rⁿ-valued stochastic process X to actually be n-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion.

Let X = (X₁, ..., X_n) be a continuous stochastic process on a probability space (Ω, Σ, P) taking values in Rⁿ. Then the following are equivalent:

1. X is a Brownian motion with respect to P, i.e., the law of X with respect to P is the same as the law of an n-dimensional Brownian motion, i.e., the push-forward measure X_∗(P) is classical Wiener measure on C₀([0, ∞); Rⁿ).
2. both
   1. X is a martingale with respect to P (and its own natural filtration); and
   2. for all 1 ≤ i, j ≤ n, X_i(t) X_j(t) − δ_ij t is a martingale with respect to P (and its own natural filtration), where δ_ij denotes the Kronecker delta.

Spectral content

The spectral content of a stochastic process ${\displaystyle X_t}$ can be found from the power spectral density, formally defined as $${\displaystyle S(\omega )=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}\mathbb{E}\left\{{\left|{\int }_0^T{e}^{i\omega t}{X}_{t}dt\right|}^2\right\},}$$ where ${\displaystyle \mathbb{E}}$ stands for the expected value. The power spectral density of Brownian motion is found to be $${\displaystyle S_{BM}(\omega )=\frac{4D}{{\omega }^2}.}$$ where D is the diffusion coefficient of X_t. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e., $${\displaystyle S^{(1)}(\omega ,T)=\frac{1}{T}{\left|{\int }_0^T{e}^{i\omega t}{X}_{t}dt\right|}^2,}$$ which for an individual realization of a Brownian motion trajectory, it is found to have expected value ${\displaystyle {\mu }_{BM}(\omega ,T)}$ $${\displaystyle {\mu }_{\text{BM}}(\omega ,T)=\frac{4D}{{\omega }^2}\left[1-\frac{\sin \left(\omega T\right)}{\omega T}\right]}$$ and variance ${\displaystyle {\sigma }_{\text{BM}}^2(\omega ,T)}$ $${\displaystyle {\sigma }_S^2(f,T)=\mathbb{E}\left\{{\left(S_T^{(j)}(f)\right)}^2\right\}-{\mu }_S^2(f,T)=\frac{20D^2}{f^4}\left[1-(6-\cos \left(fT\right))\frac{2\sin \left(fT\right)}{5fT}+\frac{(17-\cos \left(2fT\right)-16\cos \left(fT\right))}{10f^2{T}^2}\right].}$$

For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density ${\displaystyle S(\omega )}$, but its coefficient of variation ${\displaystyle \gamma =\sigma /\mu }$ tends to ${\displaystyle \sqrt{5}/2}$. This implies the distribution of ${\displaystyle S^{(1)}(\omega ,T)}$ is broad even in the infinite time limit.

Riemannian manifold

Brownian motion on a sphere

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The infinitesimal generator (and hence characteristic operator) of a Brownian motion on Rⁿ is easily calculated to be ⁠1/2⁠Δ, where Δ denotes the Laplace operator. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection. This observation is useful in defining Brownian motion on an m-dimensional Riemannian manifold (M, g): a Brownian motion on M is defined to be a diffusion on M whose characteristic operator ${\displaystyle \mathcal{A}}$ in local coordinates x_i, 1 ≤ i ≤ m, is given by ⁠1/2⁠Δ_LB, where Δ_LB is the Laplace–Beltrami operator given in local coordinates by $${\displaystyle {\mathrm{\Delta }}_{\mathrm{L}\mathrm{B}}=\frac{1}{\sqrt{det(g)}}\sum _{i=1}^m\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left(\sqrt{det(g)}\sum _{j=1}^m{g}^{ij}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\right),}$$ where [g^ij] = [g_ij]⁻¹ in the sense of the inverse of a square matrix.

Narrow escape

The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.

African Americans and birth control

From Wikipedia, the free encyclopedia

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

African Americans', or Black Americans', access and use of birth control are central to many social, political, cultural and economic issues in the United States. Birth control policies in place during American slavery and the Jim Crow era highly influenced Black attitudes toward reproductive management methods. Other factors include African-American attitudes towards family, sex and reproduction, religious views, social support structures, black culture, and movements towards bodily autonomy.

Prominent historical figures and black communities have debated whether Black Americans would benefit from the use of birth control or if birth control is inherently racist and designed to reduce the Black American population.

Early sexual and reproductive violence

Before the abolition of American slavery, enslaved Black men and women were victims of legalized sexual and reproductive violence. Some sources estimate 58% of enslaved women and girls between the ages of 15 and 30 experienced sexual assault, often perpetrated by slave masters and other White men.^[2] White people saw Black women as objects used to meet the sexual needs of white men. They also experienced abuse at the hands of the wives of the White men. White men often raped and impregnated Black women to increase their enslaved population. In addition, enslaved African-American women were subject to frequent sexual exploitation through arranged marriages in order to produce children and sexually violent encounters initiated by other enslaved people. The Jezebel stereotype shifted blame onto Black women for their experiences with sexual violence and characterized them as hypersexual. This is a stereotype that has continued within modern times. Black men were publicly lynched and castrated by white people in order to assert dominance and reduce their reproductive control.

After the Emancipation Proclamation (1865) granted legalized freedom to enslaved Black people, Jim Crow laws and Black Codes perpetrated continuing sexual and reproductive abuse of Black people, particularly women. Laws against rape only protected White women and consequently failed to protect Black survivors of gang rape, genital mutilation, and lynching.

Medical abuse and Experimentation

Medical care for enslaved Black females was rare and often carried the risk of forced medical experimentation. J. Marion Sims, known as the "father of modern gynecology", victimized enslaved African-American females with his surgical experiments by not administering anesthesia. Other physicians coerced Black women into experiments, including ones developing the cesarean section and ovariotomy.

Black men were also subjected to unethical medical research, including the Tuskegee Syphilis Study. Conducted by the United States Public Health Service, the 40-year study examined the effects of untreated syphilis in Black men in Macon County, Alabama. During the study, a treatment for syphilis became available, but the subjects were misled and denied treatment. Many of the subjects' families contracted syphilis, while the men themselves either died or developed disabilities.

Contraception

Enslaved people and birth control

In resistance to sexual exploitation and enslavement, Black women resorted to their own forms of birth control, drawing upon African folk remedies. Southern physician E. M. Pendleton reported that plantation owners frequently complained about "the unnatural tendency in the African female population to destroy her offspring".

Early organizations

After the slavery era, Black women mobilized in a variety of African-American women's clubs across the nation to exercise their political beliefs. Prominent leaders such as Harriet Tubman, Frances Harper, Ida B. Wells, and Mary Church Terrell led the founding of the National Association of Colored Women's Clubs in 1896. As issues of racism, segregation, and discrimination affected life for African-Americans in post-slavery America, the NACWC and its 1,500 affiliate clubs worked to promote racial uplift with the motto of "Lifting as We Climb", aspiring to show "an ignorant and suspicious world that our aims and interests are identical with those of all good aspiring women". Along with fundraising to establish schools and community services, the NACWC endorsed the movement for birth control as part of its agenda to empower Black women and help them achieve better lives.

In 1918, the Women's Political Association of Harlem became first African-American women's club to schedule lectures on birth control. The Harlem Community Forum invited Margaret Sanger to speak in March 1923 and the Urban League asked the American Birth Control League to establish a birth control clinic in the city. In 1925, Sanger attempted to open a clinic in the nearby, predominantly Black Columbus Hill area, but the clinic only ran for three months before closing due to low attendance. Much of the African-American population was transitioning out of the neighborhood and there was a lack of engagement with community leaders. Sanger continued to push for more clinics in struggling areas. In 1932, she opened a successful clinic in Harlem with support from Black churches and an all-Black advisory council. The clinic's clientele was about half Black and half White, and almost 3,000 people visited the clinic in its first year and a half.

Support

Harriet Tubman

In July 1932, Margaret Sanger published a special issue of her magazine entitled the Birth Control Review. The issue was titled the Negro Number and called on prominent African-Americans to display why birth control was beneficial to the African-American community. Authors such as W.E.B Du Bois and George Schuyler contributed to the magazine, each stating different reasons why they believed contraception was an asset for the Black community.

Mary Church Terrell

DuBois addressed the issue of birth control as a means of empowerment for African-Americans in the article "Black Folk and Birth Control". DuBois believed that voluntary birth control could serve as a family planning and economic tool that would empower Black families to have only as many children as they could care for. He also addressed the issue of African-Americans and the belief that in order to gain a substantial amount of power, Black people needed to produce more offspring. DuBois stated, "They must learn that among human races and groups, as among vegetables, quality and not mere quantity counts."

Angela Davis

George S. Schuyler based his entire article on the idea that the viability of black offspring was more important than the overall number of children produced. Schuyler's article, "Quantity or Quality," was a critique of the idea that sheer numbers, in terms of offspring, could bring African Americans the power and equality that they were working toward. Schuyler argued that the health of the Black family, and most specifically the health of the Black woman, should be the focus of the birth control debate. The article made it clear that if Black women were able to plan their pregnancies, then there would be a chance that the infant mortality rates would decrease. Schuyler observed, "If twenty-five percent of the brown children born die at birth or in infancy because of the unhealthful and poverty-stricken conditions of the mothers and twenty-five percent more die in youth or vegetate in jails and asylums, there is instead of a gain a distinct loss."

Opposition

Contraception was not unilaterally accepted in the African-American community during the early 20th century. Birth control to some seemed like a method of population control that could be administered by the government against Black people. Marcus Garvey and Julian Lewis were both against birth control for African-Americans for this reason, though their approaches differed. Garvey, as a Black Nationalist, believed in the "power in numbers" theory when it came to how Black people would obtain power in the U.S. Garvey was also a Roman Catholic, which may have affected his viewpoint. Lewis took a more "scientific" approach to denouncing contraception.

Abortion

Abortion continues to be a highly contested topic in the African-American community with reasons that differ from those within the mainstream abortion debate. Abortion and other forms of birth control have been stigmatized within the Black community due to the traumatic history of involuntary sterilizations that many African-American women were subjected to throughout the 20th century, as well as the history of abortion and infanticide during United States slavery.

Angela Davis, a Black feminist activist and scholar, argued that Black women were not pro-abortion, but believe in abortion rights. "If ever women would enjoy the right to plan their pregnancies, legal and accessible birth control measures and abortions would have to be complemented by an end to sterilization abuse. Davis speaks on the history of abortion and infanticide in the African-American community in Women, Race and Class. Her response to Pendleton:

"Why were self-imposed abortions and reluctant acts of infanticide such common occurrences during slavery? Not because Black women had discovered solutions to their predicament, but rather because they were desperate. Abortions and infanticides were acts of desperation, motivated not by the biological birth process but by the oppressive conditions of slavery. Most of these women, no doubt, would have expressed their deepest resentment had someone hailed their abortions as a stepping stone toward freedom."

Shirley Chisholm spoke to the debate from a political perspective in 1970. Chisholm described the decriminalization of abortions as a necessary step toward the safety of women. "Experience shows that pregnant women who feel that they have compelling reasons for not having a baby, or another baby, will break the law and even worse, risk injury or death if they must get one. Abortions will not be stopped."

The Supreme Court Decision to overturn Roe V. Wade on June 24, 2022, led to economic and health concerns for all women and specifically concerns for Black women. The legalities of abortions are now decided by state and local laws, meaning some women will have to travel to obtain a legal abortion or worry of facing criminal charges. In 2019, 38% of abortions were among Black women, 33% were among White women, 21% among Hispanic women, and 7% among women of other racial and ethnic groups. The lack of access to health care is one reason why the abortion rate for women of color is higher than white women. White women were also reported to have more access to contraceptives than minorities at 69% versus Black women at 61%. In 2008, Black women accounted for over 28% of abortions annually, more than any other racial demographic: this statistic is consistent with the reality that African-American women also experience high rates of unplanned pregnancy, largely due to a lack of access to comprehensive reproductive care and access to birth control.

Organizations

As the abortion debate has continued, there has been a surge of Black pro-life groups. These organizations believe that the womb is the "most dangerous place for an African-American child." Similar to the views of Marcus Garvey, Black anti-abortionists view the abortion movement as an attack on the African-American community as a form of genocide and a push for eugenics. Planned Parenthood and the actions of founder Margaret Sanger have become a focal point for these movements, believing their efforts continue with the goal of harming Black women.

Reproductive justice movement

Reproductive justice, a framework created by the Women of African Descent for Reproductive Justice, emerged in 1994 to catalyze a national movement that prioritized the needs of marginalized women, their families, and their communities. Reproductive justice is the "human right to maintain personal bodily autonomy, have children, not have children, and parent the children we have in safe and sustainable communities." This framework transcends the argument of choice that the women's rights, reproductive rights, and reproductive health movements emphasized, as these movements represent the needs of middle class and wealthy white women. Reproductive justice deems access to comprehensive reproductive care as essential. This includes the right to and equitable access to safe abortion as well as access to contraceptives, sexual education, STI prevention and care, and other ways to support families.

Black nationalist parties

Background

Black nationalist parties in the late 1960s and early 1970s tended to view the use of contraceptives in black populations was at best, an ill-conceived public health measure, and at worst a front for a conspiracy of black genocide. For the most part, male-dominated black nationalists were opposed to the promotion of personal fertility control and protested against government-funded family planners who they viewed to be putting forth an agenda of black population control.

Much of the opposition to fertility control was sparked by the sterilization of Minnie Lee and Mary Alice in 1973. The sisters received federally funded birth control grants from the Office of Economic Opportunity (OEO). At the Montgomery Family Planning Clinic, Minnie Lee and Mary Alice, fourteen and seventeen years old at the time respectively, both underwent surgical sterilization without informed consent. Mrs Relf, their mother was unable to read and was coerced into signing parental consent forms without being able to understand the documents. Additionally, both sisters were forced by clinic staff to sign false documents indicating that they were over twenty-one. The family later filed a complaint through the Southern Poverty Law Center citing that the treatment of the sisters at the clinic was abusive and coercive because 1) neither the mother nor her daughters gave any indication of wanting to undergo surgical sterilization, 2) neither mother nor daughters met with the physician who would perform the operation before the fact, and 3) neither mother nor daughters received information about the consequences of tubal sterilization from a physician or member of the clinic staff. The Relf case prompted many other African-American, Native-American, and Latina women to come forth with similar stories of coercion. In light of this case, many black nationalist groups came to conflate any birth control movement with a larger conspiracy of black population control.

The most vocal of these black nationalist groups were the Black Panthers and the Nation of Islam. These two organizations argued that white government family planners posed a threat to the black population by offering them birth control without other health care measures, namely, preventive medicine and hospitals, pre-and postnatal care, nutritional advice, and dentistry. They argued that birth control services remained harmful without adequate solutions to health care problems related to poverty.

Additionally, other black groups and black scholars vocally criticized the targeting of poor black communities as centers for population control. Ron Walters, chairman of the department of political science at Howard University, a historically black university, was one of the most outspoken critics of population control aimed at black families. He advocated that black communities ought to be responsible for defining their own fertility programs and birth control policies. Members of the Urban League, NAACP, and the Southern Christian Leadership Conference, likewise criticized birth control programs throughout the 1960s. A particular point of contention was the lack of minority representation in Planned Parenthood.

However, as the feminist message of the right to abortion and birth control began to become more widespread and as black feminists became more vocal in advocating for birth control access, the views of many Black Nationalist parties began to adapt. By the mid-1970s, the federal government had reduced funding for fertility control and family planning programs were viewed as less favorable after Roe v. Wade, the landmark Supreme Court abortion case. Additionally, vocal criticism of federal family planning programs leads the government to refashion its rhetoric to be less targeted toward poor black communities. Given this context, groups such as the Black Panther's expanded their emphasis on total health care to include birth control and abortion when voluntarily chosen.

Black Panther Party

Since the foundation of the Black Panther Party in 1966, the organization rejected all forms of reproductive control, claiming that governmentally regulated reproductive control was genocidal for Black people. The Black Panthers and the Black Liberation Army, the military wing of the party, believed that armed Black revolution against White supremacy was possible. They saw targeted birth control as part of a governmental plot to reduce the number of Black people in the United States, to prevent such a revolution.

Their suspicious view of birth control changed throughout the 1970s. In 1971, women in the party pushed back against an anti-birth control position on the basis that large families are difficult to support. They argued that this difficulty would make it harder for both men and women to participate politically. In 1974. Elaine Brown took over leadership of the party and actively placed other female members in leadership positions. The FBI's crackdown in the Black Power movement in the 1970s led to the arrest and/or death of many male party leaders, further increasing the influence of women in the party.

Though male party leadership was reasserted by founder Huey P. Newton in 1976, Brown's tenure as leader of the party from 1974 to 1976 significantly changed the party's stance on birth control policies and other feminist causes. In particular, the party educated Black women on the dangers of forced sterilization and published articles on documented cases of coerced sterilization by the state. In an article published by the committee to End Sterilization Abuse, the Black Panthers asserted that as high as 20% of Black women in the United States had been sterilized. Additionally, the party shifted their rhetoric to emphasize the importance of health care and legal abortion in black communities.

Nation of Islam

The Nation of Islam, a black political and religious movement founded in the 1930s, was among the first to claim that fertility control was a form of genocide. In the 1960s, the group drew parallels between what they saw as genocidal population control in the United States and population control policies in third-world countries. While they maintained a hardline approach to birth control and abortion, the group also pushed for expanded health care for black communities and greater structural solutions to health problems linked to poverty.

Sterilization

Early 1900s

The widespread practice of female sterilization began in the early 1900s. Throughout the 20th century, a majority of states passed laws allowing sterilization, and even requiring it in prescribed circumstances. The first sterilization statutes were passed in Indiana in 1907, and the last was passed in Georgia in 1970. Indiana's law allowed the "prevention of the procreation of, confirmed criminals, idiots, imbeciles, and rapists". The Supreme Court of Indiana declared this statute unconstitutional in 1921, but a similar law passed in 1927 was ruled constitutional. Over the next fifty years, laws resembling Indiana's were passed in 30 different states. These policies legalized forced sterilization for certain groups based on race and class, many of which were already marginalized. The landmark Supreme Court case Buck v. Bell (1927) upheld a state's right to forcibly sterilize a person considered unfit to procreate.

As early scientific genetic theories were emerging, eugenics (and thus sterilization) became an accepted way of protecting society from the offspring of those individuals deemed lesser than or dangerous to society—the poor, the disabled, the mentally ill, and particularly, people of color. Several states, most notably North Carolina, set up Eugenics Boards during this time period to review petitions from government and private agencies to perform sterilizations on poor, unwed, disabled women. The most popular form of female sterilization was tubal ligation, a surgical procedure that severs or seals a woman's fallopian tubes, permanently preventing her from conceiving a child. In most cases, medical providers did not have to ask for the women's consent before performing the procedure.

Mid-century practices

Sterilization abuse of black women peaked in the 1950s and 1960s.

In 1970, black women were sterilized at over twice the rate of white women: 9 per 1,000 for black women as compared to 4.1 per 1,000 for white women. A second survey taken in 1973 indicated that 43% of women who underwent sterilization in federally financed family planning programs were also black. The intersection between race and low-income status made black women even more vulnerable to forced sterilization. Many of these black women were poor and could only rely on federally subsidized clinics or Medicaid for health care. Some women had experienced sterilization without consent after a doctor had agreed to perform an illegal abortion; others were pressured into allowing sterilization after receiving a legal hospital abortion. The patterns were even worse for non-married black women; as of 1978, such women were 529 percent more likely to receive tubal sterilization than their white counterparts.

During the 1960s and 1970s, punitive sterilization laws were proposed in California, Connecticut, Delaware, Georgia, Illinois, Iowa, Louisiana, Maryland, Mississippi, Ohio, South Carolina, Tennessee, and Virginia. The purpose of such laws was to reduce the number of children born to poor, unmarried mothers. Many of these laws contained statutes that withheld welfare benefits from women with illegitimate children. As intended, these laws disproportionately affected women of color, particularly African-American mothers. According to the ACLU, the Eugenics Board of North Carolina approved 1,620 sterilizations between 1960 and 1968. Of that number, 1023 were performed on black women and nearly 56 percent of those were performed on black women under 20 years of age.

In addition to government actions, abuses of power also took place in the American medical establishment. One of the most infamous examples of such abuses was the actions of South Carolina physician, Clovis Pierce. After accepting federal money to perform the sterilizations of 18 Medicaid patients in his clinic, he told women that he would only deliver their third pregnancy on the condition that they would submit to sterilization immediately afterwards. Pierce successfully defended himself against all lawsuits, and the South Carolina branch of the American Medical Association (AMA) unanimously supported Pierce's actions.

Relf v. Weinberger

Main article: The Relf Sisters

In 1973, one particular case brought attention to the issue of forced sterilization. Twelve-year-old Minnie Lee Relf was sterilized without her consent or consent of a parent in a federally funded Department of Health, Education, and Welfare (HEW) health clinic—the Montgomery Family Planning Clinic—in Montgomery, Alabama. The official plaintiff was Lee's sister Katie Relf, and the defendant was Caspar Weinberger, secretary of the department.

The case challenged a state eugenics statute that authorized the procedure for "mentally incompetent" individuals without requiring the individual's consent or that of a guardian. Caseworkers had diagnosed Lee as mentally retarded and thus were able to apply the statute, although the basis for their diagnosis was extremely questionable. As a result, both Lee and another sister, Mary Alice, received tubal ligations without their consent. Their mother, who was illiterate, unknowingly authorized the procedure by signing an "X", under the false impression that her daughters were receiving routine birth control injections.

The District Court decided in favor of the Relf sisters, declared certain HEW regulations covering sterilizations to be "arbitrary and unreasonable", and prohibited HEW from providing federal funds for the sterilization of "certain incompetent persons". The District Court also ordered HEW to amend its overall regulations. During the course of the litigation, HEW withdrew the challenged regulations. The Court of Appeals held that the case was rendered inconsequential by HEW's actions and remanded the case back to the District Court for dismissal.

Anti-sterilization efforts

The issue of forced sterilization came to the forefront of activists' and scholars' minds in the 1960s and 1970s when evidence of widespread sterilization abuse on women of color was uncovered. Disproportionate numbers of black women, among other minority groups like Puerto Ricans and Native Americans, were receiving sterilizations, and many were completed in federally-funded clinics. In the 20th century, 32 states had federally-funded sterilization programs in place.

Anti-sterilization efforts in the 1970s came as the result of several high-profile sterilization abuse scandals. In 1972, President Nixon failed to enact HEW sterilization regulations that would have ended forced sterilization at federally-subsidized. This failure was exposed in 1973 due to Relf v. Weinberger. It was later revealed that between 100,000 and 150,000 poor women had been sterilized using federal dollars. Regulations were quickly put in place after the National Welfare Rights Organization sued HEW in 1974. These regulations "prohibited the sterilization of anyone less than 21 years of age, required a 72-hour waiting period, and protected a woman from losing her Aid to Families with Dependent Children (AFDC) support if she did not agree to sterilization". HEW, however, then created a program through which states were reimbursed for sterilizations of poor women.

The committee to End Sterilization Abuse (CESA) was founded in 1974 to combat abusive sterilization of women of color. It had a strong "anti-imperialist orientation" that attracted a multiracial membership, including white, Puerto Rican, and black women. In 1975, a coalition of groups formed an umbrella organization called the Advisory Committee on Sterilization. Members of the coalition included CESA and the National Black Feminist Organization, a group committed to addressing the double burden of racism and sexism faced by black women. The coalition formed to advise the New York City Health and Hospital's Corporation (HHC) on how to prevent forced or coerced sterilization within municipal hospitals.

The Advisory Committee created a set of guidelines that mandated a 30-day waiting period and required:

that consent not be given at the time of abortion or childbirth; that there be counseling on other fertility control options; that information on sterilization be given in the patient's native language; that the idea for sterilization must originate with the patient; that women could bring a patient advocate and another person of their choosing to accompany them through the process; and that the patient present written understanding of sterilization with an emphasis on its permanence.

In 1975, these guidelines were passed by HHC and enforced within municipal New York City hospitals, and in a 1977 City Council vote they were extended to all NYC hospitals. In 1978, the Nadler bill, named for state assemblyman Jerrold Nadler, was passed in the New York state legislature which outlined a set of sterilization regulations similar to the original ones passed by the HHC.

CESA disbanded after the passage of the Nadler bill, and with the HEW sterilization regulations in place, anti-sterilization efforts took the back burner for many feminist activists, though contemporary groups such as Incite! and SisterSong continue to address sterilization within the black community. In 2013, North Carolina became the first state to compensate its victims of forced sterilization with a payment of $50,000. In North Carolina, 7,600 people were sterilized between 1929 and 1974, 85% of them female and 40% of them nonwhite. Virginia became the second state to provide payments in 2015, giving each living victim $25,000.

Organizations

In the 1980s and 1990s, black women active in mainstream white-led reproductive rights organizations founded their own organizations, such as the National Black Women's Health Project and African American Women Evolving.

National Black Women's Health Project (NBWHP)

The first National Conference on Black Women's Health Issues was held at Spelman College in 1983. The conference led to the foundation of the National Black Women's Health Project (NBWHP) with the intent of bringing African-American women's voices on health and reproductive rights to national and international attention.

Founded by health care activist Byllye Avery and health educator Lillie Allen, and incorporated as a nonprofit organization in 1984, the NBWHP was the first reproductive justice organization for women of color. The organization changed its name in 2003 to the Black Women's Health Imperative "to reinforce the need to move beyond merely documenting the health inequities that exist for Black women and to focus on actionable steps to eliminate them".

Based in Atlanta, Georgia, the NBWHP had established chapters in 22 states by the end of 1989 and popularized its message through conferences, workshops, and publications. The NBWHP participated in the 1985 United Nations World Conference for Women in Nairobi, Kenya. In 1990, NBWHP opened a public policy–based office in Washington, D.C. to more actively promote public policies that improve black women's health, such as advising President Clinton's abortion rights policies and regulation. By then, the NBWHP had moved from being a grassroots organization to advertising black women's health through public policy on a national stage.

African American Women Evolving (AAWE)

African American Women Evolving (AAWE) started in 1996 as a project within the Chicago Abortion Fund, a predominantly white abortion rights organization. AAWE was committed to holistic community health education and promoting black women's health. After organizing Chicago's first conference on black women's health, AAWE became an independent organization in 1999, incorporating under the National Network of Abortion Funds. The organization later changed its name to Black Women for Reproductive Justice (BWRJ). BWRJ conducted several surveys on African-American women's reproductive health and made that data publicly available. A predominantly grassroots organization, BWRJ emphasized making health care information and options available to African-American women in Illinois. The organization also worked on policy recommendations and advised, inter alia, the National Abortion Reproductive Rights Action League (NARAL).