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Stochastic state-transition-change process and particle physics

The description of particle phenomena is important for basically all fields of physics. The contemporary theoretical statistical descriptions of a particle phenomenon typically requires knowledge of the Hamiltonian of the system, cross sections or other information concerning various microscopic processes. However, this knowledge may not be available and need to be determined first on the basis of experimental data. Moreover, the diverse hitherto theoretical approaches are either restricted to relatively very small class of particle phenomena or are not easily usable for analysis of experimental data. Therefore, new state-transition-change (STC) stochastic process is proposed. STC stochastic process satisfies Markov property and opens up new possibilities for obtaining in unified way deeper insight to experiments having independent outcomes. With its help it is possible to determine probability density functions characterizing states of a system and probabilities of transitions of the states on the basis of experimental data. E.g., in the context of particle physics it allows to describe in unified way particle decays, random motion of particles, particle-matter interactions (such as transmission of light, laser beam, through various optical elements), as well as particle-particle interactions (collisions). Not all particle experiments satisfy the assumptions of STC stochastic process, but broad class of particle experiments can be analyzed with its help.

Understanding of particle phenomena is essential as they concern, in one way or another, basically every field of physics. Several theoretical approaches have been developed to describe particle transport phenomena. Often an equation describing a transport phenomena originally introduced in one field of physics has been later used in another area of physics due to similarities of many transport phenomena [1,2].
Theoretical descriptions of transport phenomena may be divided into two groups: macroscopic and microscopic, each has its own advantages and disadvantages. The first group tries to describe observed transport phenomena on macroscopic level. E.g., continuum mechanics is based on the concept of continuum rather than on discrete particles (corpuscles) to model certain type of transport phenomena. The concept of continuum is used in fluid mechanics to derive, e.g., the Navier-Stokes equations to characterize motion of vari-ous viscous fluid substances. Fluid mechanics is based on the concept of momentum transfer [3]. Several transport equations used to describe momentum, heat and mass transfer are summarized in [3]. The equation of radiative transfer deals with transfer of energy in a medium which absorbs, emits or scatters radiation [4] (for applications in astrophysics see also [5]). The radiative transfer and heat transfer are specific cases of energy transfer. There are other transport equations modeling transport phenomena as an overall macroscopic effect using the concept of, e.g., momentum, energy or mass flow taking into account transport of individual particles only partially or not at all.
The second group of transport theories 1 is based on the concept of particles. Time dependent position and momentum of one particle can be described with the help of Newton's second law of motion (law of force) or Hamilton's equations. These equations describe motion of particles deterministically. Let us focus on statistical descriptions of particle transport phenomena.
Consider a group of N particles. Hamiltonian formulation of mechanics allows to derive the Liouville equation. The equation merges classical mechanics with probability theory, and it represents one of the basic equations of statistical mechanics. The Liouville equation is the evolution equation of the probability density function of finding particle at given position, momentum and time. It is necessary to know both the Hamiltonian of the system and initial position and momentum of each of the particles to determine the probability density function. Determining the probability density function may be, therefore, very difficult or not possible at all in many cases (especially if N is high).
The Liouville equation can be reduced, under certain conditions, to the Boltzmann equation also called Boltzmann transport equation [8]. It has been introduced in thermodynamics by L. Boltzmann in 1872; it has, therefore, long history. Using the Boltzmann equation one may obtain less detailed statistical description of the physical system than using the Liouville equation, but one does not need to know the initial positions and momenta of all the particles (the number of degrees of freedoms is significantly reduced). The Boltzmann equation requires to know the external force acting on the particles. The interactions between particles, if not neglected, are described by a collision term which is in general not easy to determine. Analytical solution of the Boltzmann equation is known only in simpler cases; various numerical methods have been developed in the past to find (at least approximate) solution of the Boltzmann equation in more complicated cases [8,9]. 1 The term kinetic theory [6] is sometimes [7] regarded as the aspect of statistical mechanics that is concerned with the derivation and study of equations for the particle phase space density or distribution function (called kinetic equations or transport equations). The term transport theory is in [7] restricted to mathematical discipline concerned with the solutions of kinetic equations and applications of the solutions to the study of particle transport processes. We will not use this distinction. The term kinetic theory will not be used at all and the term transport theory will be used in very general sense denoting any theory which describes transport phenomena, not necessary related directly to particles, and try to provide some insight into transport processes.
The Boltzmann equation or similar equations (including relativistic and quantum mechanical analogues) are widely used for statistical description of many transport phenomena. Theoretical description of charged particle transport in gases and condensed matter is summarized in detail in [10]. The equation forms a basis of molecular thermodynamics [11]. Electron transport in semiconductors is described with the help of the Boltzmann equation, too [12]. The Vlasov-Fokker-Planck equation, special form of the Boltzmann equation, found its application in plasma modeling [13][14][15].
Another widely known equation which is closely connected to statistics and probabilities is the Schroedinger equation. It requires knowledge of the Hamiltonian of the system (or the Hamiltonian needs to be determined on the basis of experimental data). Its application to particle transport phenomena may be, however, very delicate already in the case of simple physical phenomena such as (elastic) proton-proton scattering at high energies (more comments to description of this collision process will be given later in sect. V D).
Monte Carlo (MC) methods represent another broad group of theoretical statistical methods used for description of particle transport phenomena. To simulate a random process, i.e., sample from a probability density function for an event, it uses random numbers or pseudorandom numbers. MC methods are summarized in [16]. To make a MC simulation one first needs to identify basic physical processes. Determination of the probability density functions of the processes typically requires detailed knowledge of cross sections of various interaction types, mean free paths, spatial distribution and types of scattering centers, etc.
It is worth to mention statistical optics [17]. It deals with statistical properties of light waves. Its application to data concerning propagation of light phenomena is based on several strong assumptions (such as introduction of complex amplitudes, various approximations concerning the amplitudes etc.). It does not fully explore the concept of quantum of light -photon (discovered by M. Planck [18,19] and A. Einstein [20] at the beginning of 20th century), and it can be hardly used for other particle types and in different fields than optics.
The contemporary theoretical statistical method used for description of particle transport phenomena have something in common. They all require knowledge of the Hamiltonian of the studied system, various cross sections or other information which, however, in many cases is not a priory known and need to be obtained first, using a suitable method, on the basis of experimental data. E.g., a particle from a beam passes through a medium with certain probability and, if it passes through, its properties may change according to the corresponding conditional probability density function. The contemporary widely used theoretical statistical methods (see also the statistical mechanics [21]) do not allow to easily determine these conditional probabilities. Moreover, e.g., the probability density function determined on the basis of the Boltzmann equation is function of position, momentum and time (i.e., 7 variables in total); however, different variables may be essential for explanation of measured characteristics of given particle transport phenomenon.
Particle transport may be regarded as stochastic (random) process, a system is transition through some states described by random variables. Brownian motion (random walk) is a well known example of stochastic process. Another well known stochastic process used in context of particle physics (decay of particles) is Poisson process. In this paper the theory of stochastic processes will be used to extend and unify description of particle transport phenomena. This paper is structured as follows. The statement of the problems is further discussed in sect. II. Newly introduced state-transition-change (STC) stochastic process is introduced in sect. III where several formulae describing the process are derived, too. General guidelines for analysis of experimental data using the probabilistic model are in sect. IV. Several examples of applications of the stochastic process to experimental data of particle phenomena are summarized in sect. V. Sects. V B to V D show how four very different types of particle phenomena (particle decay, particle motion, particlematter interaction and particle-particle interaction) can be all described with the help of the STC process. Concluding remarks can be found in sect. VI.

II. PROBLEM STATEMENT
Definition II.1 (particles). By particles we mean any countable set of object such as photons, neutrons, electrons, ions, atoms, molecules, etc.
Definition II.2. Control surface A control surface is called two dimensional surface (flat or curved) which one can imagine anywhere in space. It may or may not correspond to physical surface of some matter objects or media (such as prisms, foils, glass, water, mirror, ...). Let us consider a particle beam passing through a sequence of control surfaces Σ i (i ∈ (0, . . . , M) as schematically shown in fig. 1. Particles can have very different properties, see definition II.1. The control surfaces (see definition II.2) may or may not be orthogonal to the beam. The control surfaces define M transport segments as shown in fig. 1. i-th transport segment (i ∈ (0, . . . , M − 1)) may contain any medium, field or free space (i.e., anything) between two control surfaces Σ i and Σ i+1 .
There are many particle transport phenomena which have this sequential arrangement or more general arrangement containing splitting and merging of beams as it is very common in many laser (optical) systems. In the case of light beams there are many optical components widely used to change properties of the beam (prisms, polarizers, Faraday rotators, lenses, beam splitters, mirrors, various electro-optic and magnetooptic devices, etc.). In the case of charged particle beams various magnets and other devices used for beam steering are commonly denoted as optical components, too.
Interaction of a particle with i-th transport segment may depend on various variables (parameters) characterizing the transport segment, e.g., rotation of an optical component (its spatial orientation with respect to the beam), an electric or magnetic field applied to it, temperature of a medium, etc. These variables may depend, e.g., on time or other variables.
Various detector technologies and experimental methods are used to measure quantities characterizing both beam particles passing through the control surfaces Σ i and properties of a medium placed between Σ i and Σ i+1 . These quantities are typically measured as functions of only some of the variables characterizing state of system consisting of particles and transport segment, in limited intervals of their values and sometimes the quantities are measured without the normalization. I.e., it is often difficult or even impossible to determine all characteristics of the system on experimental ground.
More experimental information about the system (particle beam and transport segments) may be obtained by changing the order of the transport segments in the sequence and measuring the quantities for each permutation of the transport segments. A lot of useful experiments data about given transport segment ("sample") may be obtained if it is irradiated by beams of different properties and the beam properties of the output (reflected or transmitted) beams are measured. The transport segments which precede the given sample modify the properties of the input source beam. The transport segments following the sample in the sequence allow better analysis of the properties of the output beam which interacted with the sample.
As to the theoretical description of particle transport phenomena. As mentioned in the introduction there are several theoretical approaches used for description of particle transport phenomena but they have limited region of applicability. They are used typically only for a specific class of particle transport phenomena and provide only partial insight. Unified description is missing. They are many particle transport phenomena which have never been satisfactorily explained at all. Transmission of particles through a medium is often considered as probabilistic process but the probability (density) functions characterizing the transitions have never been determined (except very special cases). Transmission of light trough 3 linear polarizers may serve as an example which has small number of optical elements, the transmission of light can be measured as function of rotation angles of the polarizers, but its contemporary theoretical description is not satisfactory.

III. PROBABILITY MODEL
Sect. III A summarizes for convenience basic relations and definitions concerning probability and probability density functions. It also introduces basic notation which will be used later. Control event defined in sect. III A 6 and density of states of an event defined in sect. III A 7 are the only terms which are not widely used, or may have different meaning.
Sect. III B starts with standard definitions concerning stochastic processes [22][23][24]. The general concept of density of states of an event is then used to derive several relations describing newly introduced stochastic process. A. Probability

Probability, basic relations
Definition III.1 (Probability). The probability P(A) of an event A is the proportion (relative frequency) of times (denoted as N(A)) that the event is expected to occur when an experiment is repeated a large number of times N under identical conditions Definition III.2 (Conditional probability). Let A and B be two events. If P(A) = 0 then the conditional probability of A given B is defined as the ratio where P(A) is the unconditional probability of A, and P(A∩B) is the probability that both events A and B occur (∩ denotes the intersection). It holds (for the reason of symmetry) Proposition III.1. Let A 0 , A 1 , . . . , A n−1 where n ≥ 2 be a sequence of events. If P(A 0 ) = 0 then Proof. Eq. (2) implies the chain rule (n ≥ 2) Using eq. (5) recursively one obtains This allows to factorize the probability P(A 0 ∩ ... ∩ A n−1 ) with the help of conditional probabilities. By changing the order of the events one obtains different expressions how the probability P(A 0 ∩ ... ∩ A n−1 ) can be factorized. The factorization of the probability can be chosen according to given use case. It holds (see eq. (2)) Using eqs. (5) and (7) one obtains Taking into account the assumption that P(A 0 ) = 0 and dividing eqs. (6) and (7) by P(A 0 ) one obtains eq. (4).
Remark III.1. Given two events A and B, and B can be expressed as union of n >= 2 events, then proposition III.1 allows factorization of conditional probability P(B | A) into a product of n probabilities.
Definition III.3. Probability P(A) of an event A can depend on one or more variables. In this case one can write P A (. . . ) where ". . . " denotes the variables (arguments of the functions). Similar notation can be introduced for conditional probability and other functions of an event (given possibly another event).

Integration over multiple variables
Let x represents several variables, i.e., let x = (x 0 , x 1 , . . . , x n−1 ) be finite ordered set of n variables. Multiple integral of a function f (x) over its domain may be then written as .. If . . , (x n−1 ) 1 ) and x 2 = ((x 0 ) 2 , (x 1 ) 2 , . . . , (x n−1 ) 2 ) then one may define If a variable has discrete values then the corresponding integration needs to be replaced by sum.
We may further define an abbreviation for n-dimensional interval and infinitesimal interval with a convention that for a discrete variable the corresponding interval contains only one value.
Remark III.2. A simple rule for understanding an expression (operation) containing variables represented by x is the following. One may first imagine that x represents only one continuous variable. If the variable is discrete then the expression needs to be replaced by analog expression for discrete variable (replacing an integral by sum, infinitesimally small interval by a point, ...). If x represents more than one variable then the expression needs to be generalized for multiple variables. If n = 0 then the integration of a function over x may be removed from the expression. Integration theory provides deeper reasoning of these rules, see chapter 2 in [24].
Remark III.3. In the context of probability theory functions which are typically non-negative (being product of probability functions and probability density functions having finite normalization) are often integrated over many variables. The Fubini-Tonelli theorem can be used to change of the order of integrations (and existence of the multidimensional integral).

Probability (density) function
Definition III.4. Let us denote the non-random variables specifying an event A as X NR (A) and the random variables of A as X(A).
Definition III.5. Infinitesimal interval of continuous random variables of an event A is denoted as dX(A) or, equivalently, as d(A), i.e., If the event A has no continuous random variable then Definition III.6. If X(A) contains at least one continuous random variable then the probability P(A) can be expressed with the help of probability density function ρ P(A) is infinitesimally small probability (it will not be denoted as dP(A)).
Definition III.7. In the case of conditional probability of event A given event B one can introduce conditional probability density function ρ then the probability density function is normalized to unity Proof. By inserting eq. (19) to eq. (21) one obtains eq. (22).
Remark III.4. In the case of probabilities depending on continuous random variables one can replace symbol P in equation like eq. (2), containing multiplication of probabilities) by ρ (resp. d) to obtain analogical expressions for probability density functions (resp. infinitesimal intervals). Probability density functions may be factorized similarly as one can factorize probability functions depending on discrete random variables, see sect. III A 1. E.g., it holds (see eq. (8)) and (if ρ(A 0 ) = 0, see proposition III.1)

Mutually independent events
Definition III.8 (Mutually independent events). Two events A and B are mutually independent if and only if it holds If two events are not mutually independent then they are mutually dependent. More generally, a family {A n } n∈N of events is called mutually independent if for any finite set of indices Remark III.5. The word mutually is often omitted for the sake of simplicity.

Control event
Assumption III.1. Given set (event) is not specified by any random continuous variable.
Proposition III.5. If an event B satisfies assumption III.1 then it holds Proof. It follows from definition III.5.
Definition III.9 (Control set (event)). A control set (event) B is an event satisfying assumption III.1.

Density of states
Definition III. 10. Density of states represented by event A is the ratio Proposition III.8. Let B be a control set and A a subset of B. It holds Proof. It follows from eqs. (35) and (38) (see definition III.9 and proposition III.7).
Remark III.6. The eq. (39) can be used as alternative definition of density of states corresponding to event A if another event B satisfying eq. (25) is given.
Proposition III.9. Integration of dos(A) over all its random variables given a control event B (which may reduce the limits of integration) yields number of occurrences of the control event B Proof. Integration of eq. (39) (see proposition III.8) over X(A | B) and using eq. (33) (see proposition III.6) imply eq. (40).
Remark III.7. N(B) represents normalization of the density dos(A) according to proposition III.9. Integration of dos(A) over just some random variables of A (given control event B) one obtains density of event corresponding to the random variables over which it is not integrated.
Remark III.8. Many quantities like particle number flux, number density (number of specified objects per unit volume), density of energy states of a system, etc. are commonly used in physics to describe density of states of various physical systems. Definition III.10 represents unification and generalization of these quantities used in physics in special cases.
Proposition III. 10. Let B be a control set. If N(B) = 0 then Proof. If N(B) = 0 then one can divide eq. (39) by N(B) being equal to the right-hand side of eq. (40).
B. Stochastic STC processes

Generalities
Definition III.11 (Index set). Let I be an index set with the Borel σ -algebra I (the smallest σ -algebra on I containing all open sets or, equivalently, closed sets).
Definition III.12 (Probability space). A probability space is triple (Ω, F , P), where 1. Ω is a sample space, i.e., the set of all possible outcomes (also called realizations).
2. F is an event space, i.e., σ -algebra on Ω (admissible subsets of Ω, a subset called event being a set of outcomes in the sample space).
3. P is a probability function which assigns each event in the event space a probability, a number between 0 and 1.
Definition III.13 (Stochastic process). A stochastic process is a family X = {X i : i ∈ I} of random variables defined on the same probability space (Ω, F , P) and, for fixed index i, taking their values X i in given space S i which must be measurable with respect to some σ -algebra S i of admissible subsets.
Definition III.14 (States and state space). The values of random variables X i for fixed i ∈ I represent "states" and the set of all states corresponding to all the possible values of the random variables X i is called "state space" S i of the stochastic process for fixed i.
Remark III.9 (Different state spaces). Many stochastic processes are often a priory restricted to one common state space for all i ∈ I. Definition III.13 is more general. In definition III.13 it is allowed that state spaces S i can be different for different indexes i ∈ I. Moreover, a one-to-one correspondence of the states from one state space to another state space may or may not exist. If a state s i transitioned to a state s j (i ∈ I, j ∈ I, i = j), then the values of the random variables X i may or may not differ from the values of random variables X j . Moreover, there may or may not be different numbers of variables in X i and X j , or the numbers may be the same but their meaning may be completely different, or some of the random variables may or may not have the same domains.
Definition III.15 (Non-random variables). Let X NR i be nonrandom variables (parameters) influencing whether a system will be in a state s i in S i for given index i ∈ I.
Remark III.10 (Dependence of functions on non-random variables). Values of random variables X i and non-random variables X NR i (parameters) specify a state s i in state space S i (and vice versa). A system can transition from one state to another state. Therefore, a function of state s i may in general depend also on all the non-random variables further specifying states of the system before they transitioned to a state in state space S i . A function describing a transition of a subset of S i to a subset of S j may in addition depend on non-random variables X NR j . The dependence on the non-random parameters may or may not be written explicitly (as needed).
Definition III.16 (Sample function). For each ω ∈ Ω, the function i ∈ I → X i (ω i ) ∈ S i is called the sample function, a realization or outcome (sometimes called also ω-trajectory, or ω-sample path), of the stochastic process {X i : i ∈ I}. ω i is one of the states s i ∈ S i . Definition III.17 (Branching process). Stochastic process {X i : i ∈ I} is called branching process if it holds for all indexes i ∈ I: if a state s i in state space S i transition to another state space S j (i ∈ I, j ∈ I, i = j) then there is no other state space from which a state could transition to the state space S j .
Remark III.11. State space S i is a control set (event), see definition III.9. Density of states introduced in sect. III A 7 is, therefore, suitable for quantifying densities of states s i ∈ S i . Definition III.18. Let us define (the newly defined functions being on the left-hand sides) for all i ∈ I.
Proposition III.12. Let {X i : i ∈ I} be stochastic process given by definition III. 13.
for all i ∈ I. Proof.

STC process
Assumption III.2 (Probabilistic transition of states). A state s i from state space S i (i ∈ I) may or may not transition to another state or states being in one or more state spaces.
Remark III.12. Assumption III.2 can be reduced to assumptions concerning stochastic process where state spaces S i are the same for all i ∈ I and where a state of system always transition to another state (in this case the transition probability is equal to 1). The assumption III.2 is, therefore, very general.
Remark III.13. Assumption III.3 is a strong requirement. There is, of course, significant loss of generality when description of stochastic processes is limited by this assumption.
On the other side, the assumed independence strongly simplifies description of stochastic processes. Many processes (not only) in physics satisfy this assumption.
In the example of transport of particles discussed in sect. II, the assumption III.3 excludes phenomena when, e.g., a bunch of particles is absorbed by a medium such that transmission of any other particle through the medium is influenced by it. E.g., the absorbed particles may change the properties of the medium. This is the case of, e.g., stimulated emission of photons where probabilities of transmission of two photons through a medium may differ in dependence on number of already absorbed photons in the medium.
Remark III.14. Assumption III.3 does not imply that, e.g., if a state s i 1 transition to state s i 2 and state s i 2 to state s i 3 (i 1 = i 2 = i 3 , i 1 ∈ I, i 2 ∈ I, i 3 ∈ I) then the conditional probabilities of the two transitions are not necessarily independent (they may or may not be independent).
Assumption III.4 (Markov property). Let {X i : i ∈ I} be stochastic process given by definition III.13. Let the process have Markov property. E.g., if index set I is totally ordered ≤ and (F i , i ∈ I) is filtration then for each i 2 , i 1 ∈ I with i 1 < i 2 .
Remark III.15. Roughly speaking, assumption III.4 means that probability of future state of system depends on its present state, but not on the past states in which the system was. This property is also known as memoryless. Some processes are Markovian and some are not. The Markov property can appear to be violated when, e.g., not all important variables are not taken into account in description of a system (i.e., if states of the system are not sufficiently specified). The Markov property can appear to be violated also due to insufficient amount of data (information about the system). Assumption III.4 cannot be made without a loss of generality. Markovian processes are subject of research with long history, they have found applications in many fields. Remark III.16. If N 0 = 0 then there is no initial state in S 0 which could transition to S 1 , etc. Therefore, one can assume N 0 = 0 without lost of generality. If N i = 0 for a fixed i which satisfies 0 < i ≤ M then N j = 0 for all j ∈ I for which i ≤ j ≤ M (i.e., the system never transition to a state s j ).
Proposition III.13. Let {X i : i ∈ I} be stochastic process given by definition III.21. If N 0 = 0 then for all i ∈ I.
Definition III.22. Let {X i : i ∈ I} be stochastic process given by definition III.21. One can introduce the following functions (being on the left-hand sides of the equations) Proposition III.14. Let {X i : i ∈ I} be stochastic process given by definition III.21. It holds Proof. It holds Eqs. (54) and (56) and Proof. Eqs. (57) and (58) or, equivalently, (using different notation, see eqs. (45), (51) and (52)) Proof. The probability P(s i ∩ s i+1 | S i ) can be factorized into 3 conditional probabilities (see assumptions III.2 to III.4 and the chain rule given by eq. (4)) The factorization allows to derive several important formulae. Two out of the three conditional probabilities on the righthand side of eq. (61) can be infinitesimal probabilities (if they are specified by at least one random continuous variable); they may be expressed with the help of the corresponding probability density functions ρ: Eq. (62) can be rewritten in different notation (writing explicitly the arguments of the functions) Integrating eq. (63) over X i , multiplying it by N S i and using eq. (53) (see proposition III.14) yields For the purpose of using proposition III.1 one can introduce events A 0 = S i , A 1 = S i+1 and A 2 = s i+1 . S i = / 0 implies P(A 0 ) = 0 (see definition III.21). Given process is a sequence and it holds S i = / 0 and S i+1 = / 0 according to the assumptions. It implies A 1 = A 1 ∩ A 0 . The chain rule given by proposition III.1 for probability P( Multiplying eq. (65) by N S i and using eq. (57) gives Eqs. (64) and (66) imply With the help of definition of density of states, see eq. (39), one can write Eqs. (68) to (70) imply eq. (59).
Remark III.17. The key step to proof theorem III.1 is to factorize the probability P(s i ∩ s i+1 | S i ) into 3 terms, see eq. (61). This equation is then basically only integrated over all the random variables characterizing the input states, and several basic relations concerning probabilities are used together with the definition of density of states.
Remark III. 18. The factorization of the probability P(s i ∩ s i+1 | S i ) into 3 terms is one of the key ideas of the whole STC process. The first term corresponds to conditional probability of system being in a fixed input state given that it is in i-th state space (here the concept of density of states is essential). The second term corresponds to the conditional probability of transition of the state to (i + 1)-th state space given that the system was in the state. The third term corresponds to the conditional probability of change of the input state to given output state given that the input state transitioned to (i + 1). Experimental data may constrain differently the individual probabilities, i.e., functions dos i (X i ), P T,i (X i ) and ρ C,i (X i , X i+1 ). The factorization simplify taking into account additional knowledge (assumptions) about the evolving system. The factorization of the probability P(s i ∩s i+1 | S i ) is one of the key ideas of simplifying and unifying descriptions of various systems corresponding to assumptions III.2 to III.4). It is also the motivation for calling the stochastic process state-transition-change (STC) process.
Remark III.19. Theorem III.1 is of major importance as it shows how to transform density of states dos s i (X i ) (characterizing the input states) to density of states dos s i+1 (X i+1 ) (characterizing the output states).
Remark III.20. The right-hand side of eq. (72) does not depend on the value of N 0 . Eq. (71), therefore, implies that the relative density of states dos i (X i )/N 0 does not depend on the value of N 0 for any i >= 0. When the input probability density function ρ S , 0 (X 0 ) in dependence on values of X 0 corresponding to the state space S 0 is given (see eq. (72)) then eq. (71) represents recursive formula for obtaining the relative densities of events dos i (X i )/N 0 (i = 1, . . . , M).
Corollary III.1. Let {X i : i ∈ I} be stochastic process given by definition III.21.
Remark III.21. It means that the relative number of events N i+1 /N 0 given by corollary III.1 does not depend on function ρ C,i (X i , X i+1 ) characterizing the i-th transition; it, however, depends on functions ρ C, j (X j , X j+1 ) for j = 0, . . . , i−1 as they are needed to calculate dos j (X j )/N 0 for any j = 0, . . . , i − 1 with the help of proposition III. 16. It also means that it is not possible to determine ρ C,i (X i , X i+1 )| i=M−1 corresponding to the last (M − 1)-th transition on the basis of measured ratio N i /N 0 , even if the ratios are known for all i = 0, . . . , M (i.e., including the last state space S M ).
Proposition III.17. Let {X i : i ∈ I} be stochastic process given by definition III.21. If N 0 = 0 then Proof. No variable represented by X i is random variable of the conditional probability (density) functions ρ s i+1 |s i ∩S i+1 (X i , X i+1 ) and P s i+1 |s i ∩S i+1 (X i , X i+1 ) given event s i ∩ S i+1 . Only random variables X i+1 are random variables of these functions. If N 0 = 0 for event s i ∩ S i+1 then Eqs. (22) and (76) Proof. Integration of eq. (59) over X i+1 (see theorem III.1), propositions III.9 and III.18 imply eq. (77).
Proposition III. 19. Let {X i : i ∈ I} be stochastic process given by definition III. 21. The probability that system transitioned to any of the states s i+1 in S i+1 given it was in any of the states s i in S i is Proof. It holds Integrating eq. (63) over X i+1 and X i (under assumptions III.2 to III.4) and using eqs. (46) and (79) It holds if and only if If the probability P S i+1 |s i (X i ) < 1 then Integration of this inequality over X i and eq. (46) imply This inequality can hold if and only if P S i+1 |s i (X i ) < 1.
The equality holds if and only of P S i+1 |s i (X i ) = 1. Remark III.22. Propositions III. 19 and III.20 show that some "probability density functions" may not be normalized to unity (compare it to, e.g., eq. (46) and proposition III.18 where the probability density functions are normalized to unity). This is consequence of assumption III.2 and that given process is a sequence, i.e., a branching process.
Definition III.23 (STC sequence -case 1). Let {X i : i ∈ I} be stochastic process given by definition III.21. Let it satisfy assumption III.5.
Definition III.24. The probability density function ρ C,i (X i , X i+1 ) satisfying the assumption III.5 and corresponding to the i-th transition may be denoted as ρ C,i (X i+1 ). It holds (i = 0, . . . , M − 1)
Definition III.25 (STC sequence -case 2). Let us define stochastic process {X i : i ∈ I} as in definition III.23 and let it satisfy also assumption III.6.
Definition III.26. The probability density function ρ C,i (X i , X i+1 ) satisfying both the assumptions III.5 and III.6 may be denoted as ρ C and it holds in this case i.e., (see eqs. (44) and (51)) Proof. Definition III.10 and assumption III.7 imply that the densities of states s i and s i+1 are It holds It holds according to corollary III.2 and eq. (99) (under assumptions III.2 to III.4) X i does not represent any random variable, therefore, there is effectively no integration in eq. (101). Eqs. (100) and (101) imply eq. (97).
Remark III.24. The proof of proposition III.21 shows how the simple and expected result represented by eq. (98) is derived under special conditions from much more general corollary III.2.

C. Place-transition net of (stochastic) processes
Remark III.25. Dynamics of a system can be described with the help of a state-transition net, also called place-transition (PT) or Petri net. A PT net is directed bipartite graph that has two types of elements: places and transitions. One can define control sets (see definition III.9) and some quantities characterizing states of the system in the control sets (e.g., density of states corresponding to each of the control sets, see definition III.10). The quantities can represent places and transformation of the quantities represent the transitions between different places. The transitions can be specified with the help of theorem III.1, corollary III.2, and proposition III.21; or some other transformations if needed.
Various types of PT nets are studied in mathematics and computer science; one can take advantage of the achieved results and developed tools. Description of dynamics of a system using a PT net substantially simplifies modeling (simulation) of the system if it consists of many places and transitions between them. Modeling of a system using a PT net provides higher level of abstraction and it helps to visualize complexity of the places and transitions.
Remark III.26. As to the example in sect. II it is possible to define transformations corresponding to splitting of a beam as well as merging of two or more beams into one. Description of the splitting (using a beam splitter) is relatively straightforward. It is necessary to define transformation of, e.g., density of states corresponding to a control set before the split and a control set corresponding to given split beam (it can be done for all the split beams individually, the control sets must be disjoint). If particle beam can only split and do not merge then the stochastic process is branching process defined by definition III.17. Description of merging of two or more beams is more delicate to describe, but it can be done, too. Laser systems (having typically many components and including concurrent processes) can be, therefore, represented by a PT net.

IV. GENERAL GUIDELINES FOR ANALYSIS OF EXPERIMENTAL DATA
Analysis of experimental data using a probabilistic model can be divided into several steps (being analogous to an analysis of data using any other model). We will focus mainly on using STC process.

A. Collection of data and model formulation
Gathering various measured data concerning a system and its initial inspection is the first step in formulation of a (probabilistic) model. Measured data often point to symmetries, even without analysing them in detail using sophisticated theoretical methods. The symmetries typically simplify formulation of the model. One may define a place-transition net as described in sect. III C. In the case of a stochastic STC process it is necessary to define state spaces S i and both random X i and non-random X NR i variables characterizing states s i ∈ S i for all indexes i in an index set I. On the basis of experimental data it must be possible to determine the number of states in state spaces S i or density of states in state spaces S i as a function of the variables (the more detailed experimental information is available the better).
The initial inspection of data helps also in determination of properties of the system which can be added to the model as additional assumptions (see, e.g., STC sequence given by definition III.23 and sect. III B 6). The additional assumptions allows to derive more specialised formulae and perform various calculations analytically. Formulation of the model depends also on its purpose (e.g., introducing only limited number of variables typically does not allow to fully describe the system, but it may allow to determine some partial characteristics of the system, which cannot be measured directly, very efficiently if only essential variables are taken into account).

B. Check of applicability of the probabilistic model to measured data (consistency)
Before making a sophisticated analysis of data using the model it is useful to make basic consistency checks. E.g., it may be possible to test experimentally the basic assumptions III.3 and III.4 of a STC process (see definition III. 19). In the case of a STC sequence (see definition III.21) one can test eq. (74) (i.e., one can test assumption III.2).
If these conditions are not satisfied then at least one of the assumptions of the model must be changed to remove the contradiction. On the other hand, if these conditions are satisfied then one may continue to analyse experimental data with the help of the probabilistic model.

C. Parameterizations of unknown functions
All the unknown function in the model can be parameterized. E.g., STC sequence of M transitions (see definition III.21) requires in the most general case to parameterize density of states dos i (X i ) for all i ∈ (0, . . . , M), and probability P T,i (X i ) and ρ C,i (X i , X i+1 ) for all i ∈ (0, . . . , M − 1).

D. Fitting of a probabilistic model to data
When one tries to find the values of all the free parameters of the model using an optimization procedure then basically three possibilities may occur:

No solution is found under given set of assumptions.
This can indicate a contradiction in the model (e.g., the parameterizations may exclude the possibility from the very beginning independently on the values of the free parameters, or at least one of the assumptions of the model is not correct); or it may be just very difficult optimization problem.
2. More than one solution is found.

Only one solution is found.
This must be taken into account in making correct conclusions from an analysis of experimental data with the help of the model. In some cases it may be very useful, or even necessary, to modify some of the assumptions of the model and try to determine the parameterized functions on the basis of experimental data again. This can help to better understand the whole system and make correct interpretation of the results. It is also important for proposing additional experiments to test the results. Constrained optimization methods allows to take into account various experimental information into account.

E. Separation of determination of probabilities from other additional assumptions
An analysis of experimental data with the help of a probabilistic model can be divided into two stages:

determination of probability (density) functions
One may first try to find the unknown probability (density) functions. Their parameterizations can contain many free parameters to be flexible enough. If a parameterization is a priory too restrictive it may not allow finding solution to the optimization problem (the solution may be excluded from the very beginning). While the free parameters may not have physical meaning, the probability (density) functions do have meaning and should be able to describe experimental data under given assumptions without any contradiction. The point of this analysis aim is to explore, test, and understand well the basic probabilistic character of an evolving system.

determination of physical mechanisms (causes)
One may try to find physical processes (causes) which could lead to the probability (density) functions which can be (in one way or another) fitted to experimental data. The free parameters of these physical processes may be fitted to the determined probability (density) functions. E.g., in the case of transmission of particles through matter (see sect. II) one may try to express the determined probability (density) functions with the help of cross sections of various interactions as the free parameters, and try to find their values by fitting them to the already determined probability functions. In some cases, depending on the studied physical system, it may be possible to similarly study, e.g., the Hamiltonian of the system, forces responsible for transition of one state of the system to another. This analysis aim tries to describe the system in greater detail, but it requires deeper insight to the studied system.
The possibility of the separation of the whole analysis of experimental data with the help of an probabilistic model into two stages is essential for understanding various phenomena (especially those which are more complex). Determining first the unknown probability (density) functions and then trying to explain the physical mechanisms may be in many cases much more effective than trying to guess all the key physical processes (some of them may not be even known), calculating the corresponding probability (density) functions, and then trying to fit the free parameters using experimental data. On the other hand in some (simpler) cases good guess of the physical mechanisms may help to find the probability (density) functions describing the measured data much faster. In one way or another it is advantageous to study and understand how much each of the probability (density) functions is constrained by data and how much by some additional assumptions. This allows to tests different assumptions and interpretation possibilities of experimental data using falsification approach (see sect. VII.A in [25] for further details). Many theoretical statistical approaches, see sect. I, do not easily allow division of the whole analysis of data into the two stages (they are often based on some strong assumptions from the very beginning).

F. Computational complexity
As to the computational complexity. To determine the parameterized probability (density) functions on the basis of experimental data it may be necessary to calculate (many) integrals, see, e.g., one of the main formulae characterizing STC sequence given by eq. (60). The number is increasing with increasing number of variables. The integrals can be calculated analytically only in relatively simple cases, in the other cases they need to be calculated numerically which can be very time consuming. The computational complexity can be significantly reduced when the integrals are simplified under some additional assumptions, see eq. (96) as an example. where is probability of one particle not decaying in the time interval from t 0 to t, and λ is positive constant called the decay constant.
Proof 1. It follows from assumptions V.1 and V.2 that the number of decayed particles at time t is where the minus sign expresses the decrease of the undecayed particles (λ being positive constant). Eq. (109) can be rearranged Integrating both sides (left side in the limits from N(t 0 ) to N(t), and right side in the corresponding limits t 0 , t) implies Exponentiating both sides gives eq. (107).
Proof 2. Let {X i : i ∈ I} be stochastic process given by definition V.2. It holds (constant λ will not be written explicitly as an argument of a function of number of particles or a probability function in the following) The conditional probability P T,i (t i ,t i+1 ) does not depend on parameters t 0 , . . . ,t i−1 given that the particle have not decayed until time t i . Assumption V.2 implies that probability P ∆ that a particle does not decay in time interval (t,t + ∆t) for fixed ∆t > 0 does not depend on time, and the probability P T,i (t i ,t i+1 ) is the same for all i = 0, . . . , M − 1 (see eq. (104)). It holds The probability P ∆ can be approximated by The smaller ∆t the better the assumption V.2 is satisfied. It holds in the limit M → ∞ and using eqs. (103) and (113) to (115) one obtains Eqs. (102) and (116) imply eq. (108).
Remark V.1. Eq. (107) can be used for analysis of experimental data to determine the free parameter of the decay process, i.e., the decay constant λ .
Remark V.2. The proof 1 is standard proof which can be found in textbooks. The proof 2 demonstrates that the exponential decay can be derived with the help of STC process introduced in sect. III B. The proof 2 also shows one way how one can use STS process and deal with continuous transitions (see the limit M → ∞). STC process can help to study decay processes in more complicated cases when, e.g., more variables may be needed to describe the decay products, i.e., the output states (input and output states may differ, see remark III.9). Or, experimental data do not correspond to the exponential formulae given by eq. (108) (not even in more general case when more decay constants corresponding to different types of particles in the sample are considered).
Remark V.3. Corollary V.1 can be, and it is standardly done, proven in yet another way, using Poisson process defined in chapter 5 in [22]. Different approaches in derivation of one formula help to better study the involved assumptions which can or cannot lead to the same results.

B. Motion of particles
Motion of particles (bodies) in presence of an external force field can be, in some cases, described with the help of Newton's second law of motion. I.e., as deterministic process, if the initial conditions of the particles are precisely known. If the initial conditions are characterized by a probability density function then the process can be considered as random (stochastic).
As an example one may consider a source emitting particles of different speeds and at different times in presence of no external force as an example of nontrivial particle motion. It may not be possible to detect the particles at the place where they are emitted but it may be possible to measure some time dependent characteristics characterizing the particle transport at a distance from the source. The characteristics of the emitted particles can be determined on the basis of time dependent quantities which can be measured at different distances from the source. This is a subject of well known time resolved velocity spectrometry (a type of time-of-flight method). It is shown in [26] under which assumptions STC process is reduced to the formulae of time resolved velocity spectrometry.
The results from [26] can be generalized. It is possible to introduce non-zero external force and other properties of particles (such as electric charge, mass, ...) which are characterized by corresponding probability density functions (not known precisely). Transport of particles may depend on these particle characteristics. This allows to study forces acting on particles having some properties specified by random variables. Inertial mass increase in dependence on velocity may be also studied on the basis of experimental data under these conditions [27].
STC stochastic process, therefore, opens up new possibilities of determination of particle characteristics (and forces) in situations where randomness of the particle motion must be taken into account.
C. Particle-matter interaction

Transmission of light through polarizers
The probabilistic model developed in sect. III has been used in [28] for description of transmission of photon beam through 3 linear polarizers. In this example of particle-matter interaction each photon from the beam has been characterized by one variable, one degree of freedom characterizing direction of polarization and having meaning of an angle. Each polarizer in the sequence has been characterized by its rotation angle. One example of input data of ratio of number of transmitted photons to initial number of photons in front of and behind each of the 3 polarizers in dependence on their rotation angles has been analysed in [28]. The probability (density) functions dos i (X i ), P T,i (X i ) and ρ C,i (X i , X i+1 ) (being function of both random and non-random variables) charactering interaction of the photon beam with the polarizers have been determined with the help of the formulae concerning STC sequence discussed in sects. III B 3 to III B 5.
This example of description of 3 polarizer experiment shows how the probabilistic character of transmission of light through other optical elements may be studied with the help of a (STC) stochastic process. This may be useful, e.g., in further development of various components used in laser systems (see sect. II). The probabilistic model, therefore, opens up new possibilities how to study polarization of light and particlematter interactions in general.

Particle beam as diagnostic tool
Various medical as well as industrial radiographic methods use ionizing and non-ionizing radiation for diagnostic purposes of various samples. Sample of different states of matter (gaseous, solid, liquid or in even more exotic) are widely studied.
E.g., laser is essential tool in chemistry, see [29] where one can find many widely used experimental methods for measuring properties of a particle beam (laser) before and after an interaction with a sample (which can be basically arbitrary). STC stochastic process can help to extend possibilities of analyses of experimental data.
Let us mention also some regions where particle beams are used as diagnostic tools but where description of the system is delicate, with or without (STC) stochastic process. These concern mainly systems which are highly dynamical (involve many time dependent processes), non-linear (having complicated properties depending on time and other variables), not exactly reproducible (initial conditions may vary), nonhomogeneous and exist only for short amount of time (this makes high demands on their measurement). Limited experimental statistical information is often available for these system.
A short-lived localized plasma sample (quasi-neutral system of electrons and ions) may be produced in laboratory when high-power laser beam is focused onto a solid target. The created plasma sample may be irradiated by another (diagnostic) laser beam and the beam intensity of the transmitted diagnostic beam may be measured (e.g., by a CCD camera), as it was demonstrated at Prague Asterix Laser System (PALS) in [30]. The measurements performed in [30] shows how polarometry and interferometry can be combined to obtain important experimental information about a plasma sample. The generated plasma is not homogeneous and, therefore, the intensity of diagnostic beam (particle flux) is measured in dependence on transversal coordinates.
Another plasma diagnostic method using a particle beam is ion deflectometry, see [31] (and, e.g., [32][33][34]). Ions carrying electric charge and traversing plasma sample interact very differently than neutral photons. It means that ion deflectometry may provide information about a plasma sample which cannot be determined with the help of the photon diagnostic beam (and vice versa).
Better understanding of, e.g., laser plasma interaction and various other related particle transport phenomena is important for many areas of physics including the context of inertial confinement fusion [35]. Sample of warm dense matter (WDM) can be prepared and diagnosed in laboratory similarly as plasma sample, see [36] (a WDM sample is similarly dynamical physical system as the plasma sample).
STC stochastic process can be useful in studies of these "exotic" samples mainly by helping to determine properties of the diagnostic beams before and after their interactions with the beam, see for example sect. V C 1; and also in determination of properties of particles emitted from the samples, see for example sect. V B.
Description of the processes inside a sample caused by particles from a beam will often require dedicated models which do not correspond to the assumptions of STC process. This applies not only to processes inside the samples mentioned above, but also to modeling of, e.g., radiobiological mechanisms standardly divided into two consecutive stages: physical-chemical and biological [37]. In [37] Petri nets were used to describe chemical stage (combined effect of diffusion of cluster of radicals and chemical reactions). Probabilistic two-stage model (see sect. 6.3. in [37]) allows to describe response of irradiated cell to damage caused by radiation. This example demonstrates that place-transition (Petri) nets, theory of probability make possible to study and model efficiently complex system divided into very diverse stages. Theory of stochastic processes can further extend possibilities of this approach in various areas of scientific research, see also sect. III C.

D. Particle-particle interaction
The probabilistic description of particle transport discussed in sect. III is useful also for description of particle-particle collisions.
Number of particle-particle collisions corresponding to an event is equal to cross-section of the event times luminosity. Concept of luminosity (for both fixed target and colliding beams experiments) is closely related to density of states of the beam(s) (i.e., particle densities), see [38]. Probabilistic theory is also used, e.g., for calculation of pileup probabilities for specific events per bunch-crossing as discussed in [39]. If the luminosity and the number of collisions corresponding to an event is measured then the cross section of the event can be determined experimentally.
E.g., in [40] one may find example of one of many measurements of elastic proton-proton differential cross section performed by TOTEM 2 experiment at the Large Hadron Collider (LHC) at CERN at the center-of-mass energy of 8 TeV. The measurement of scattered protons moving close to the beam requires dedicated detector technology and experimental techniques, see [41] for details and future plans extending TOTEM physics program.
However, theoretical understanding of the measured crosssections is in many cases very delicate. Possibilities, limitations and open questions in contemporary models of elastic collisions of particles interacting strongly at short distances (hadrons) have been summarized in [42]. Theoretical description of elastic hadron collisions is more delicate than than it may seem at first glance, even though it is kinematically the simplest collision process and a lot of elastic scattering data are available at various collision energies, see [43][44][45] for further details.
To overcome the difficulties summarized in [42] a probabilistic particle collision model has been developed and used for description of measured elastic proton-proton differential cross section at energy of 52.8 GeV and some preliminary results have been obtained, see sect. 6 in [46]. With the help of the model it has been possible to take into account, at given collision energy, density of input two particle states distinguished by impact parameter. It allowed to determine probabilities of total, elastic and inelastic strong (short-ranged) interaction in dependence on the values of impact parameter and sizes of the colliding particles. A final state of two particles scattered elastically has been characterized by (mean) elastic scattering angle as a function of impact parameter; the model has allowed to determine this dependence, too.
STC stochastic process introduced in sect. III B can provide deeper insight into the probabilistic collision model introduced in sect. 6 in [46] and also to the concept of luminosity. STC stochastic process is also essential for developing further the probabilistic collision model. E.g., measured cross-sections of various types of interactions (not only elastic differential cross sections) can be analyzed consistently with its help, too.

VI. CONCLUSION
Applicability of contemporary statistical descriptions of particle phenomena, see sect. I, for analysis of experimental data is in many cases restricted to relatively small class of phenomena, based on several strong assumptions, requires Hamiltonian of the system or other information which may not be known and may be difficult to determine the theoretical methods.
State-transition-change stochastic process has been introduced in sect. III B to describe various systems transitioning through sequence of states. Main assumptions limiting applicability of the process to data are assumptions III.3 and III.4, i.e., Markov property and independence of outcomes. Theorem III.1 represents key theorem simplifying description of these systems. With the help of STC process it is possible to formulate a probability model of various phenomena taking into account both available experimental data and addition assumptions (constrains). The approach allows separation of analysis of experimental data into two stages. In the first stage, one may try to determine the probability (density) functions characterizing evolution and states of the system. In the second stage, one may try to explain the established probability (density) functions on the basis of microscopic processes, as it has been discussed in sect. IV. Moreover, the model may depend only on variables which are essential for description of given phenomenon. The possibility of separation of the whole analysis of data into two stages and using only essential variables for given purpose is essential for deeper understanding of many particle phenomena as it allows to test various assumptions (interpretation possibilities) effectively.
Analysis of experimental data using STC process combines probability theory, theory of stochastic processes, infinitesimal calculus and optimization theory. Place-transition (Petri) nets can be used to simplify description of more complex systems (possibly with the help of other theoretical models, based on different assumptions than the assumptions of STC process). STC process, therefore, extends possibilities of contemporary descriptions of many systems. Stochastic STC process is distinct from other stochastic processes already used in physics [47,48]; it, therefore, broaden applicability of stochastic theory in physics.
The various examples discussed in sect. V show that STC process can unify description of many phenomena, even if they may look very distinct at first glance. E.g., with its help it is possible to describe both motion of particles having initial values described with the help of probability density function, see sect. V B and [26], and transmission of light through various optical elements, see sect. V C 1 and [28]. The unified description of many particle phenomena is not only possible but also highly desirable in terms of developing an intrinsic understanding of these phenomena which are important, in one way or another, for all areas of physics.