Freud's psychoanalytic theories

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Sigmund Freud (c. 1921)

Sigmund Freud (6 May 1856 – 23 September 1939) is considered to be the founder of the psychodynamic approach to psychology, which looks to unconscious drives to explain human behavior. Freud believed that the mind is responsible for both conscious and unconscious decisions that it makes on the basis of psychological drives. The id, ego, and super-ego are three aspects of the mind Freud believed to comprise a person's personality. Freud believed people are "simply actors in the drama of [their] own minds, pushed by desire, pulled by coincidence. Underneath the surface, our personalities represent the power struggle going on deep within us".

Views on religion

Further information: Sigmund Freud's views on religion

Freud did not believe in the existence of a supernatural force that has pre-programmed us to behave in a certain way. His idea of the Id explains why people act out in certain ways when it is not in line with the ego or superego. "Religion is an illusion and it derives its strength from the fact that it falls in with our instinctual desires." Freud believed that people rely on religion to give explanations for anxieties and tension they do not want to consciously believe in. Freud argued that humanity created God in their image. This reverses the idea of any type of religion because he believed that it is constructed by the mind. The role of the mind is something that Freud repeatedly talked about because he believed that the mind is responsible for both conscious and unconscious decisions based on drives and forces. The idea that religion causes people to behave in a moral way is incorrect according to Freud because he believed that no other force has the power to control the ways in which people act. Unconscious desires motivate people to act accordingly.

Freud did a significant amount of research studying how people act and interact in a group setting. He believed that people act in different ways according to the demands and constraints of the group as a whole. In his book Group Psychology and the Analysis of the Ego, Freud argued that the church and organized religion form an "artificial group" which requires an external force to keep it together. In this type of group, everything is dependent on that external force, and without it, the group would no longer exist. Groups are necessary, according to Freud in order to decrease narcissism in all people, by creating libidinal ties with others by placing everyone at an equal level. The commonness among different people with different egos allows people to identify with one another. This relates to the idea of religion because Freud believed that people created religion in order to create these group ties that they unconsciously seek.

Oedipus complex

According to Freud's many theories of religion, the Oedipus complex is utilized in the understanding and mastery of religious beliefs. In Freud's psychosexual stages, he mentioned the Oedipus complex and how it affects children and their relationships with their same-sex parental figure. According to Freud, there is an unconscious desire for one's mother to be a virgin and for one's father to be an all-powerful, almighty figure. Freud's interest in Greek mythology and religion greatly influenced his psychological theories. The Oedipus complex is when a boy is jealous of his father. The boy strives to possess his mother and ultimately replace his father as a means of no longer having to fight for her undivided attention and affection. Along with seeking his mother's love, boys also experience castration anxiety which is the fear of losing their genitalia. Boys fear that their fathers will retaliate and castrate them as a result of desiring their mother. Females also experience penis envy which is the parallel reaction to the male experience of castration anxiety. Females are jealous of their fathers' penis and wish to have one as well. Girls then repress this feeling and instead long for a child of their own. This suppression leads to the girl identifying with her mother and acquiring feminine traits.

Psychoanalytic theory

Main article: Id, ego and super-ego

Psychoanalysis was founded by Sigmund Freud. Freud believed that people could be cured by making their motivations conscious. The aim of psychoanalysis therapy is to release repressed emotions and experiences, i.e., make the unconscious conscious. Psychoanalysis is commonly used to treat depression and anxiety disorders. It is only by having a cathartic (i.e., healing) experience that a person can be helped and "cured".

Id

The Id, according to Freud, is the part of the unconscious that seeks pleasure. His idea of the Id explains why people act out in certain ways when it is not in line with the ego or superego. The Id is the part of the mind which holds all of humankind's most basic and primal instincts. It is the impulsive, unconscious part of the mind that is based on the desire to seek immediate satisfaction. The Id does not have a grasp on any form of reality or consequence. Freud understood that some people are controlled by the Id because it makes people engage in need-satisfying behavior without any regard for what is right or wrong. Freud compared the Id and the Ego to a horse and a rider. The Id is compared to the horse, which is directed and controlled by the Ego, the rider. This example goes to show that although the Id is supposed to be controlled by the Ego, they often interact with one another according to the drives of the Ego. The Id is made up of two biological instincts, Eros which is the drive to create, and Thanatos which is the drive to destroy.

Ego

In order for people to maintain a realistic sense here on earth, the Ego is responsible for creating a balance between pleasure and pain. It is impossible for all desires of the Id to be met and the Ego realizes this but continues to seek pleasure and satisfaction. Although the Ego does not know the difference between right and wrong, it is aware that not all drives can be met at a given time. The reality principle is what the Ego operates on to help satisfy the Id’s demands while also compromising with the constraints of reality. The Ego is a person's "self" composed of unconscious desires. The Ego takes into account ethical and cultural ideals in order to balance out the desires originating in the Id. Although both the Id and the Ego are unconscious, the Ego has close contact with the perceptual system. The Ego has the function of self-preservation, which is why it has the ability to control the instinctual demands from the Id.

"The ego is first and foremost a bodily ego; it is not merely a surface entity but is itself the projection of a surface. If we wish to find an anatomical analogy for it we can best identify it with the ‘cortical homunculus’ of the anatomists, which stands on its head in the cortex, sticks up its heels, faces backwards and, as we know, has its speech-area on the left-hand side. The ego is ultimately derived from bodily sensations, chiefly from those springing from the surface of the body. It may thus be regarded as a mental projection of the surface of the body, representing the superficies of the mental apparatus."

Superego

The Superego, which develops around age four or five, incorporates the morals of society. Freud believed that the Superego is what allows the mind to control its impulses that are looked down upon morally. The Superego can be considered to be the conscience of the mind because it has the ability to distinguish between reality as well as what is right or wrong. Without the Superego, Freud believed people would act out with aggression and other immoral behaviors because the mind would have no way of understanding the difference between right and wrong. The Superego is considered to be the "consciousness" of a person's personality and can override the drives from the Id. Freud separates the Superego into two separate categories; the ideal self and the conscience. The conscience contains ideals and morals that exist within a society that prevent people from acting out based on their internal desires. The ideal self contains images of how people ought to behave according to society's ideals.

The unconscious

Main article: Unconscious mind

Freud believed that the answers to what controlled daily actions resided in the unconscious mind, despite alternative views that all our behaviors were conscious. He felt that religion is an illusion based on human values that are created by the mind to overcome inner psychological conflict. He believed that notions of the unconsciousness and gaps in the consciousness can be explained by acts of which the consciousness affords no evidence. The unconscious mind positions itself in every aspect of life, whether one is dormant or awake. Though one may be unaware of the impact of the unconscious mind, it influences the actions we engage in. Human behavior may be understood by searching for an analysis of mental processes. This explanation gives significance to verbal slips and dreams. They are caused by hidden reasons in the mind displayed in concealed forms. Verbal slips of the unconscious mind are referred to as a Freudian slip. This is a term to explain a spoken mistake derived from the unconscious mind. Traumatizing information on thoughts and beliefs is blocked from the conscious mind. Slips expose our true thoughts stored in the unconscious.

Sexual instincts or drives have deeply hidden roots in the unconscious mind. Instincts act by giving vitality and enthusiasm to the mind through meaning and purpose. The range of instincts is in great numbers. Freud expressed them in two categories. One is Eros, the self-preserving life instinct containing all erotic pleasures. While Eros is used for basic survival, the living instinct alone cannot explain all behavior, according to Freud. In contrast, Thanatos is the death instinct. It is full of self-destruction of sexual energy and our unconscious desire to die. The main part of human behavior and actions is tied back to sexual drives. Since birth, the existence of sexual drives can be recognized as one of the most important incentives of life.

Psychosexual stages

Main article: Psychosexual development

Freud's theory of psychosexual development is represented by five stages. According to Freud, each stage occurs within a specific time frame of one's life. If one becomes fixated in any of the five stages, he or she will develop personality traits that coincide with the specific stage and its focus.

• Oral Stage – The first stage is the oral stage. An infant is in this stage from birth to eighteen months of age. The main focus in the oral stage is pleasure-seeking through the infant's mouth. During this stage, the need for tasting and sucking becomes prominent in producing pleasure. Oral stimulation is crucial during this stage; if the infant's needs are not met during this time frame he or she will be fixated in the oral stage. Fixation in this stage can lead to adult habits such as thumb-sucking, smoking, over-eating, and nail-biting. Personality traits can also develop during adulthood that is linked to oral fixation; these traits can include optimism and independence or pessimism and hostility.
• Anal Stage – The second stage is the anal stage which lasts from eighteen months to three years of age. During this stage, the infant's pleasure-seeking centers are located in the bowels and bladder. Parents stress toilet training and bowel control during this time period. Fixation in the anal stage can lead to anal-retention or anal-expulsion. Anal retentive characteristics include being overly neat, precise, and orderly while being anal expulsive involves being disorganized, messy, and destructive.
• Phallic Stage – The third stage is the phallic stage. It begins at the age of three and continues until the age of six. Now, sensitivity becomes concentrated in the genitals and masturbation (in both sexes) becomes a new source of pleasure. The child becomes aware of anatomical sex differences, which sets in motion the conflict of jealousy and fear which Freud called the Oedipus complex (in boys). Later, the Freud scholars added the Electra complex (in girls).
• Latency Stage – The fourth stage is the latency stage which begins at the age of six and continues until the age of eleven. During this stage, there is no pleasure-seeking region of the body; instead, all sexual feelings are repressed. Thus, children are able to develop social skills and find comfort through peer and family interaction.
• Genital Stage – The final stage of psychosexual development is the genital stage. This stage starts from eleven onwards, lasts through puberty, and ends when one reaches adulthood at the age of eighteen. The onset of puberty reflects a strong interest from one person to another of the opposite sex. If one does not experience fixation in any of the psychosexual stages, once he or she has reached the genital stage, he or she will grow into a well-balanced human being.

Anxiety and defense mechanisms

Freud proposed a set of defense mechanisms in one's body. This set of defense mechanisms occurs so that one can hold a favorable or preferred view of oneself. For example, in a particular situation when an event occurs that violates one's preferred view of oneself, Freud stated that it is necessary for the self to have some mechanism to defend itself against this unfavorable event; this is known as a defense mechanism. Freud's work on defense mechanisms focused on how the ego defends itself against internal events or impulses, which are regarded as unacceptable to one's ego. These defense mechanisms are used to handle the conflict between the id, the ego, and the superego.

Freud noted that a major drive for people is the reduction of tension and the major cause of tension is anxiety. He identified three types of anxiety; reality anxiety, neurotic anxiety, and moral anxiety. Reality anxiety is the most basic form of anxiety and is based on the ego. It is typically based on the fear of real and possible events, for example, being bitten by a dog or falling off a roof. Neurotic anxiety comes from an unconscious fear that the basic impulses of the id will take control of the person, leading to eventual punishment for expressing the id's desires. Moral anxiety comes from the superego. It appears in the form of a fear of violating values or moral codes and appears as feelings like guilt or shame.

When anxiety occurs, the mind's first response is to seek rational ways of escaping the situation by increasing problem-solving efforts and a range of defense mechanisms may be triggered. These are ways that the ego develops to help deal with the id and the superego. Defense mechanisms often appear unconsciously and tend to distort or falsify reality. When the distortion of reality occurs, there is a change in perception which allows for a lessening in anxiety, resulting in a reduction of the tension one experiences. Sigmund Freud noted a number of ego defenses that were noted throughout his work, but his daughter, Anna Freud, developed and elaborated on them. The defense mechanisms are as follows: 1) Denial is believing that what is true is actually false 2) Displacement is taking out impulses on a less threatening target 3) Intellectualization is avoiding unacceptable emotions by focusing on the intellectual aspects 4) Projection is attributing uncomfortable feelings to others 5) Rationalization is creating false but believable justifications 6) Reaction Formation is taking the opposite belief because the true belief causes anxiety 7) Regression is going back to a previous stage of development 8) Repression is pushing uncomfortable thoughts out of conscious awareness 9) Suppression is consciously forcing unwanted thoughts out of our awareness 10) Sublimation is redirecting ‘wrong’ urges into socially acceptable actions. These defenses are not under our conscious control and our unconscious will use one or more to protect ourselves from stressful situations. They are natural and normal and without these, neurosis develops, such as anxiety states, phobias, obsessions, or hysteria.

Totem and Taboo

Totem and Taboo

Freud desired to understand religion and spirituality and dealt with the nature of religious beliefs in many of his books and essays. He regarded God as an illusion, based on the infantile need for a powerful father figure. Freud believed that religion was an expression of underlying psychological neuroses and distress. In some of his writing, he suggested that religion is an attempt to control the Oedipal complex, as he goes on to discuss in his book Totem and Taboo.

In 1913, Freud published the book, Totem and Taboo. This book was an attempt to reconstruct the birth and the process of development of religion as a social institution. He wanted to demonstrate how the study of psychoanalysis is important in the understanding of the growth of civilization. This book is about how the Oedipus complex, which is when an infant develops an attachment for the mother early on in life, and the incest taboo came into being and why they are present in all human societies. The incest taboo rises because of a desire for incest. The purpose of the totemic animal is not for group unity, but to reinforce the incest taboo. The totemic animal is not a symbol of God but a symbol of the father and it is an important part of religious development. Totemism originates from the memory of an event in pre-history where the male group members eat the father figure due to a desire for the females. The guilt they feel for their actions and for the loss of a father figure leads them to prohibit incest in a new way. Totemism is a means of preventing incest and as a ritual reminder of the murder of the father. This shows that sexual desire, since there are many social prohibitions on sexual relations, is channeled through certain ritual actions and all societies adapt these rituals so that sexuality develops in approved ways. This reveals unconscious desires and their repression. Freud believes that civilization makes people unhappy because it contradicts the desire for progress, freedom, happiness, and wealth. Civilization requires the repression of drives and instincts such as sexual, aggression, and the death instinct in order that civilization can work.

According to Freud, religion originated in prehistoric collective experiences that became repressed and ritualized as totems and taboos. He stated that most, if not all religions, can be traced back to early human sacrifice including Christianity in which Christ on the cross is a symbolic representation of killing the father and eating the father figure is shown with ‘the body of Christ’, also known as Communion. In this work, Freud attributed the origin of religion to emotions such as hatred, fear, and jealousy. These emotions are directed towards the father figure in the clan from the sons who are denied sexual desires towards the females. Freud attributed totem religions to be a result of extreme emotion, rash action, and the result of guilt.

The Psychopathology of Everyday Life

The Psychopathology of Everyday Life

The Psychopathology of Everyday Life is one of the most important books in psychology. It was written by Freud in 1901, and it laid the basis for the theory of psychoanalysis. The book contains twelve chapters on forgetting things such as names, childhood memories, mistakes, clumsiness, slips of the tongue, and determinism of the unconscious. Freud believed that there were reasons that people forget things like words, names, and memories. He also believed that mistakes in speech, now referred to as a Freudian Slip, were not accidents but instead the "dynamic unconscious" revealing something meaningful.

Freud suggested that our everyday psychopathology is a minor disturbance of mental life which may quickly pass away. Freud believed all of these acts to have an important significance; the most trivial slips of the tongue or pen may reveal people's secret feelings and fantasies. Pathology is brought into everyday life which Freud pointed out through dreams, forgetfulness, and parapraxes. He used these things to make his case for the existence of an unconscious that refuses to be explained or contained by consciousness. Freud explained how the forgetting of multiple events in our everyday life can be a consequence of repression, suppression, denial, displacement, and identification. Defense mechanisms occur to protect one's ego so in The Psychopathology of Everyday Life, Freud stated, "painful memories merge into motivated forgetting which special ease". (p. 154)

Three Essays on the Theory of Sexuality

Three Essays on the Theory of Sexuality, sometimes titled Three Contributions to the Theory of Sex, written in 1905 by Sigmund Freud explores and analyzes his theory of sexuality and its presence throughout childhood. Freud's book describes three main topics in reference to sexuality: sexual perversions, childhood sexuality, and puberty. His first essay in this series is called "The Sexual Aberrations." This essay focuses on the distinction between a sexual object and a sexual aim. A sexual object is an object that one desires, while the sexual aim is the acts that one desires to perform with the object. Freud's second essay is titled "Infantile Sexuality." In this essay, he insists that children have sexual urges. The psychosexual stages are the steps a child must take in order to continue having sexual urges once adulthood is reached. The third essay Freud wrote describes "The Transformation of Puberty." In this essay, he examines how children express their sexuality throughout puberty and how sexual identity is formed during this time frame. Freud ultimately attempted to link unconscious sexual desires to conscious actions in each of his essays.

The Interpretation of Dreams

The Interpretation of Dreams is one of Sigmund Freud's best-known published works. It set the stage for his psychoanalytic work and Freud's approach to the unconscious with regard to the interpretation of dreams. During therapy sessions with patients, Freud would ask his patients to discuss what was on their minds. Frequently, the responses were directly related to a dream. As a result, Freud began to analyze dreams, believing that it gave him access to one's deepest thoughts. In addition, he was able to find links between one's current hysterical behaviors and past traumatic experiences. From these experiences, he began to write a book that was designed to help others to understand dream interpretation. In the book, he discussed his theory of the unconscious.

Freud believed that dreams were messages from the unconscious masked as wishes controlled by internal stimuli. The unconscious mind plays the most imperative role in dream interpretation. In order to remain in a state of sleep, the unconscious mind has to suppress negative thoughts and represent them in any edited form. Therefore, when one dreams, the unconscious makes an effort to deal with conflict. It would enable one to begin to act on them.

There are four steps required to convert dreams from latent or unconscious thoughts to the manifest content. They are condensation, displacement, symbolism, and secondary revision. Ideas first go through a process of condensation that takes thoughts and turns them into a single image. Then, the true emotional meaning of the dream loses its significance in an element of displacement. This is followed by symbolism representing our latent thoughts in visual form. A special focus on symbolism was emphasized in the interpretation of dreams. Our dreams are highly symbolic with an underlying principle meaning. Many of the symbolic stages focus on sexual connotations. For example, a tree branch could represent a penis. Freud believed all human behavior originated from our sexual drives and desires. In the last stage of converting dreams to manifest content, dreams are made sensible. The final product of manifest content is what we remember when we come out of our sleep.

Boltzmann equation

From Wikipedia, the free encyclopedia

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

The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book)

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element ${\displaystyle d^3\mathbf{r}}$) centered at the position ${\displaystyle \mathbf{r}}$, and has momentum nearly equal to a given momentum vector ${\displaystyle \mathbf{p}}$ (thus occupying a very small region of momentum space ${\displaystyle d^3\mathbf{p}}$), at an instant of time.

The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas). See also convection–diffusion equation.

The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.

Overview

The phase space and density function

The set of all possible positions r and momenta p is called the phase space of the system; in other words a set of three coordinates for each position coordinate x, y, z, and three more for each momentum component p_x, p_y, p_z. The entire space is 6-dimensional: a point in this space is (r, p) = (x, y, z, p_x, p_y, p_z), and each coordinate is parameterized by time t. A relevant differential element is written $${\displaystyle d^3\mathbf{r}{d}^3\mathbf{p}=dxdydzd{p}_{x}d{p}_{y}d{p}_z.}$$

Since the probability of N molecules, which all have r and p within ${\displaystyle d^3\mathbf{r}{d}^3\mathbf{p}}$, is in question, at the heart of the equation is a quantity f which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time t. This is a probability density function: f(r, p, t), defined so that, $${\displaystyle dN=f(\mathbf{r},\mathbf{p},t)d^3\mathbf{r}{d}^3\mathbf{p}}$$ is the number of molecules which all have positions lying within a volume element ${\displaystyle d^3\mathbf{r}}$ about r and momenta lying within a momentum space element ${\displaystyle d^3\mathbf{p}}$ about p, at time t. Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

$${\displaystyle \begin{array}{rl}N& =\underset{\mathrm{m}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}}{\int }{d}^3\mathbf{p}\underset{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}{\int }{d}^3\mathbf{r}f(\mathbf{r},\mathbf{p},t)\\ & =\underset{\mathrm{m}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}}{\iiint }\underset{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}{\iiint }f(x,y,z,p_x,p_y,p_z,t)dxdydzd{p}_{x}d{p}_{y}d{p}_z\end{array}}$$

which is a 6-fold integral. While f is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with deterministic many-body systems), since only one r and p is in question. It is not part of the analysis to use r₁, p₁ for particle 1, r₂, p₂ for particle 2, etc. up to r_N, p_N for particle N.

It is assumed the particles in the system are identical (so each has an identical mass m). For a mixture of more than one chemical species, one distribution is needed for each, see below.

Principal statement

The general equation can then be written as $${\displaystyle \frac{df}{dt}={\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\text{force}}+{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\text{diff}}+{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\text{coll}},}$$

where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.

Note that some authors use the particle velocity v instead of momentum p; they are related in the definition of momentum by p = mv.

The force and diffusion terms

Consider particles described by f, each experiencing an external force F not due to other particles (see the collision term for the latter treatment).

Suppose at time t some number of particles all have position r within element ${\displaystyle d^3\mathbf{r}}$ and momentum p within ${\displaystyle d^3\mathbf{p}}$. If a force F instantly acts on each particle, then at time t + Δt their position will be ${\displaystyle \mathbf{r}+\mathrm{\Delta }\mathbf{r}=\mathbf{r}+\frac{\mathbf{p}}{m}\mathrm{\Delta }t}$ and momentum p + Δp = p + FΔt. Then, in the absence of collisions, f must satisfy

$${\displaystyle f\left(\mathbf{r}+\frac{\mathbf{p}}{m}\mathrm{\Delta }t,\mathbf{p}+\mathbf{F}\mathrm{\Delta }t,t+\mathrm{\Delta }t\right)d^3\mathbf{r}{d}^3\mathbf{p}=f(\mathbf{r},\mathbf{p},t)d^3\mathbf{r}{d}^3\mathbf{p}}$$

Note that we have used the fact that the phase space volume element ${\displaystyle d^3\mathbf{r}{d}^3\mathbf{p}}$ is constant, which can be shown using Hamilton's equations (see the discussion under Liouville's theorem). However, since collisions do occur, the particle density in the phase-space volume ${\displaystyle d^3\mathbf{r}{d}^3\mathbf{p}}$ changes, so

$${\displaystyle \begin{array}{rl}d{N}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}}& ={\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}}\mathrm{\Delta }t{d}^3\mathbf{r}{d}^3\mathbf{p}\\ & =f\left(\mathbf{r}+\frac{\mathbf{p}}{m}\mathrm{\Delta }t,\mathbf{p}+\mathbf{F}\mathrm{\Delta }t,t+\mathrm{\Delta }t\right)d^3\mathbf{r}{d}^3\mathbf{p}-f(\mathbf{r},\mathbf{p},t)d^3\mathbf{r}{d}^3\mathbf{p}\\ & =\mathrm{\Delta }f{d}^3\mathbf{r}{d}^3\mathbf{p}\end{array}}$$

1

where Δf is the total change in f. Dividing (1) by ${\displaystyle d^3\mathbf{r}{d}^3\mathbf{p}\mathrm{\Delta }t}$ and taking the limits Δt → 0 and Δf → 0, we have

$${\displaystyle \frac{df}{dt}={\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}}}$$

2

The total differential of f is:

$${\displaystyle \begin{array}{rl}df& =\frac{\mathrm{\partial }f}{\mathrm{\partial }t}dt+\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}dy+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}dz\right)+\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_x}d{p}_x+\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_y}d{p}_y+\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_z}d{p}_z\right)\\ & =\frac{\mathrm{\partial }f}{\mathrm{\partial }t}dt+\mathrm{\nabla }f\cdot d\mathbf{r}+\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{p}}\cdot d\mathbf{p}\\ & =\frac{\mathrm{\partial }f}{\mathrm{\partial }t}dt+\mathrm{\nabla }f\cdot \frac{\mathbf{p}}{m}dt+\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{p}}\cdot \mathbf{F}dt\end{array}}$$

3

where ∇ is the gradient operator, · is the dot product, $${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{p}}={\hat{\mathbf{e}}}_x\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_x}+{\hat{\mathbf{e}}}_y\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_y}+{\hat{\mathbf{e}}}_z\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_z}={\mathrm{\nabla }}_{\mathbf{p}}f}$$ is a shorthand for the momentum analogue of ∇, and ê_x, ê_y, ê_z are Cartesian unit vectors.

Final statement

Dividing (3) by dt and substituting into (2) gives:

$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\frac{\mathbf{p}}{m}\cdot \mathrm{\nabla }f+\mathbf{F}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{p}}={\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}}}$$

In this context, F(r, t) is the force field acting on the particles in the fluid, and m is the mass of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation.

This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the Maxwell–Boltzmann, Fermi–Dirac or Bose–Einstein distributions.

The collision term (Stosszahlansatz) and molecular chaos

Two-body collision term

A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "Stosszahlansatz" and is also known as the "molecular chaos assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions: $${\displaystyle {\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\text{coll}}=\iint gI(g,\mathrm{\Omega })[f(\mathbf{r},{{\mathbf{p}}^{\prime }}_A,t)f(\mathbf{r},{{\mathbf{p}}^{\prime }}_B,t)-f(\mathbf{r},{\mathbf{p}}_A,t)f(\mathbf{r},{\mathbf{p}}_B,t)]d\mathrm{\Omega }{d}^3{\mathbf{p}}_B,}$$ where p_A and p_B are the momenta of any two particles (labeled as A and B for convenience) before a collision, p′_A and p′_B are the momenta after the collision, $${\displaystyle g=|{\mathbf{p}}_B-{\mathbf{p}}_A|=|{{\mathbf{p}}^{\prime }}_B-{{\mathbf{p}}^{\prime }}_A|}$$ is the magnitude of the relative momenta (see relative velocity for more on this concept), and I(g, Ω) is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the solid angle dΩ, due to the collision.

Simplifications to the collision term

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook. The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form:

$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\frac{\mathbf{p}}{m}\cdot \mathrm{\nabla }f+\mathbf{F}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{p}}=\nu (f_0-f),}$$

where ${\displaystyle \nu }$ is the molecular collision frequency, and ${\displaystyle f_0}$ is the local Maxwellian distribution function given the gas temperature at this point in space. This is also called "relaxation time approximation".

General equation (for a mixture)

For a mixture of chemical species labelled by indices i = 1, 2, 3, ..., n the equation for species i is

$${\displaystyle \frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }t}+\frac{{\mathbf{p}}_i}{m_i}\cdot \mathrm{\nabla }{f}_i+\mathbf{F}\cdot \frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{\mathbf{p}}_i}={\left(\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }t}\right)}_{\text{coll}},}$$

where f_i = f_i(r, p_i, t), and the collision term is

$${\displaystyle {\left(\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}}=\sum _{j=1}^n\iint g_{ij}{I}_{ij}(g_{ij},\mathrm{\Omega })[f_i^{\prime }{f}_j^{\prime }-f_i{f}_j]d\mathrm{\Omega }{d}^3{\mathbf{p}}^{\prime },}$$

where f′ = f′(p′_i, t), the magnitude of the relative momenta is

$${\displaystyle g_{ij}=|{\mathbf{p}}_i-{\mathbf{p}}_j|=|{\mathbf{p}}_i^{\prime }-{\mathbf{p}}_j^{\prime }|,}$$

and I_ij is the differential cross-section, as before, between particles i and j. The integration is over the momentum components in the integrand (which are labelled i and j). The sum of integrals describes the entry and exit of particles of species i in or out of the phase-space element.

Applications and extensions

Conservation equations

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. For a fluid consisting of only one kind of particle, the number density n is given by $${\displaystyle n=\int f{d}^3\mathbf{p}.}$$

The average value of any function A is $${\displaystyle ⟨A⟩=\frac{1}{n}\int Af{d}^3\mathbf{p}.}$$

Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus ${\displaystyle \mathbf{x}\mapsto x_i}$ and ${\displaystyle \mathbf{p}\mapsto p_i=m{v}_i}$, where ${\displaystyle v_i}$ is the particle velocity vector. Define ${\displaystyle A(p_i)}$ as some function of momentum ${\displaystyle p_i}$ only, whose total value is conserved in a collision. Assume also that the force ${\displaystyle F_i}$ is a function of position only, and that f is zero for ${\displaystyle p_i\to \pm \mathrm{\infty }}$. Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

$${\displaystyle \int A\frac{\mathrm{\partial }f}{\mathrm{\partial }t}{d}^3\mathbf{p}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}(n⟨A⟩),}$$

$${\displaystyle \int \frac{p_{j}A}{m}\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_j}{d}^3\mathbf{p}=\frac{1}{m}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}(n⟨A{p}_j⟩),}$$

$${\displaystyle \int A{F}_j\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_j}{d}^3\mathbf{p}=-n{F}_j⟨\frac{\mathrm{\partial }A}{\mathrm{\partial }{p}_j}⟩,}$$

$${\displaystyle \int A{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}\right)}_{\text{coll}}{d}^3\mathbf{p}={\frac{\mathrm{\partial }}{\mathrm{\partial }t}}_{\text{coll}}(n⟨A⟩)=0,}$$

where the last term is zero, since A is conserved in a collision. The values of A correspond to moments of velocity ${\displaystyle v_i}$ (and momentum ${\displaystyle p_i}$, as they are linearly dependent).

Zeroth moment

Letting ${\displaystyle A=m(v_i{)}^0=m}$, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation: $${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\rho +\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}(\rho V_j)=0,}$$ where ${\displaystyle \rho =mn}$ is the mass density, and ${\displaystyle V_i=⟨v_i⟩}$ is the average fluid velocity.

First moment

Letting ${\displaystyle A=m(v_i{)}^1=p_i}$, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:

$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}(\rho V_i)+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}(\rho V_i{V}_j+P_{ij})-n{F}_i=0,}$$

where ${\displaystyle P_{ij}=\rho ⟨(v_i-V_i)(v_j-V_j)⟩}$ is the pressure tensor (the viscous stress tensor plus the hydrostatic pressure).

Second moment

Letting ${\displaystyle A=\frac{m(v_i{)}^2}{2}=\frac{p_i{p}_i}{2m}}$, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:

$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(u+\frac{1}{2}\rho V_i{V}_i\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left(u{V}_j+\frac{1}{2}\rho V_i{V}_i{V}_j+J_{qj}+P_{ij}{V}_i\right)-n{F}_i{V}_i=0,}$$

where $u=\frac{1}{2}\rho ⟨(v_i-V_i)(v_i-V_i)⟩$ is the kinetic thermal energy density, and $J_{qi}=\frac{1}{2}\rho ⟨(v_i-V_i)(v_k-V_k)(v_k-V_k)⟩$ is the heat flux vector.

Hamiltonian mechanics

In Hamiltonian mechanics, the Boltzmann equation is often written more generally as $${\displaystyle \hat{\mathbf{L}}[f]=\mathbf{C}[f],}$$ where L is the Liouville operator (there is an inconsistent definition between the Liouville operator as defined here and the one in the article linked) describing the evolution of a phase space volume and C is the collision operator. The non-relativistic form of L is $${\displaystyle {\hat{\mathbf{L}}}_{\mathrm{N}\mathrm{R}}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\frac{\mathbf{p}}{m}\cdot \mathrm{\nabla }+\mathbf{F}\cdot \frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{p}}.}$$

Quantum theory and violation of particle number conservation

It is possible to write down relativistic quantum Boltzmann equations for relativistic quantum systems in which the number of particles is not conserved in collisions. This has several applications in physical cosmology, including the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density f. However, for a wide class of applications a well-defined generalization of f exists which is the solution of an effective Boltzmann equation that can be derived from first principles of quantum field theory.

General relativity and astronomy

The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f; in galaxies, physical collisions between the stars are very rare, and the effect of gravitational collisions can be neglected for times far longer than the age of the universe.

Its generalization in general relativity is $${\displaystyle {\hat{\mathbf{L}}}_{\mathrm{G}\mathrm{R}}[f]=p^{\alpha }\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}^{\alpha }}-{\mathrm{\Gamma }}^{\alpha }{}_{\beta \gamma }{p}^{\beta }{p}^{\gamma }\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}^{\alpha }}=C[f],}$$ where Γ^α_βγ is the Christoffel symbol of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (xⁱ, p_i) phase space as opposed to fully contravariant (xⁱ, pⁱ) phase space.

In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity. In the very dense medium formed by the primordial plasma after the Big Bang, particles are continuously created and annihilated. In such an environment quantum coherence and the spatial extension of the wavefunction can affect the dynamics, making it questionable whether the classical phase space distribution f that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of quantum field theory. This includes the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis.

Solving the equation

Exact solutions to the Boltzmann equations have been proven to exist in some cases; this analytical approach provides insight, but is not generally usable in practical problems.

Instead, numerical methods (including finite elements and lattice Boltzmann methods) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from hypersonic aerodynamics in rarefied gas flows to plasma flows. An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result.

Close to local equilibrium, solution of the Boltzmann equation can be represented by an asymptotic expansion in powers of Knudsen number (the Chapman–Enskog expansion). The first two terms of this expansion give the Euler equations and the Navier–Stokes equations. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of Hilbert's sixth problem.

Limitations and further uses of the Boltzmann equation

The Boltzmann equation is valid only under several assumptions. For instance, the particles are assumed to be pointlike, i.e. without having a finite size. There exists a generalization of the Boltzmann equation that is called the Enskog equation. The collision term is modified in Enskog equations such that particles have a finite size, for example they can be modelled as spheres having a fixed radius.

No further degrees of freedom besides translational motion are assumed for the particles. If there are internal degrees of freedom, the Boltzmann equation has to be generalized and might possess inelastic collisions.

Many real fluids like liquids or dense gases have besides the features mentioned above more complex forms of collisions, there will be not only binary, but also ternary and higher order collisions. These must be derived by using the BBGKY hierarchy.

Boltzmann-like equations are also used for the movement of cells. Since cells are composite particles that carry internal degrees of freedom, the corresponding generalized Boltzmann equations must have inelastic collision integrals. Such equations can describe invasions of cancer cells in tissue, morphogenesis, and chemotaxis-related effects.