J. Signal AM 6 (1978) 6, 421-439. – Journal for Signal and Amplification Materials, Akademie-Verlag, Berlin

The Reaction Tensors     
of the Photographic Process –     
as a prototype of causally consecutive processes

Ewald Gerth

Extended Abstract
The article describes the analytical treatment of the reaction-kinetic microprocesses in the silver halide crystals of the photographic emulsion during the exposure to light, which leads to systems of differential equations up to the second order. Such a system of reactions of zero, first, and second order is represented by the free electrons, the electron holes, and the traps in the crystal lattice. The build-up of the concentration centers for ions in a reaction chain can be treated as a system of reactions of first order. Only for such linear systems exact solution treatments hitherto had been applied using matrix algebra.
The system of differential equations of second order is formulated as a tensor equation, which is solved by iteration of the equivalent integral equation. The result is an absolutely and uniformly converging vector series, the algorithm of which may be programmed on computer.
The tensor representation of the reaction system also allows the simultaneous treatment of different, in principle, of any reaction orders.
For the analytical formulation of the photographic characteristic curve a tensorial version is proposed, which also includes the matrix version.
Although the discourse on the application of matrices and tensors is related especially to the photographic process, the functional relations pointed out there are valid likewise in other fields, e.g. physics, chemistry, biology etc., where reaction kinetics is of importance. Therefore, the treatise of the item outlined in the article reveals a certain universality, recommending the photographic microprocess in crystalline silver-halides as a prototype for generalization.

A current example of the universality and generalization of the photographic process is
the spread of epidemics into pandemics.

  •   Further applications of the generalized process theory:
  •   The functional mechanism of epidemics and pandemics
  •   Pandemic Freak Waves

    The photographic process – using matrix algebra:
    The matrix formulation of the photographic process gives a reasonable explanation for the fact that double exposures of equal light quantity E.t (E intensity, t time) yield different results if the sequence of short and of long-term partial exposures is exchanged. These double-exposure-effects are easily formulated by the non-commutative multiplication of matrices, which was defended at the Technical University Dresden on April 26th, 1972.
    A more comprehensive explanation of the non-commutativity of double exposures was given in an excerpt of the postdoctoral thesis by the author and in a monograph published in 2013:
    Book: Analytical representation of the kinetics of condensation-nucleus build-up in the photographic process (Fulltext in German), ISBN 978-3-8316-4299-1, Herbert Utz Verlag GmbH, Munich, 2013

    Reaction kinetics in photography – using tensor algebra:
    The treatment of kinetic processes with matrices enables a linear solution of the problem. Concerning the photographic process, however, already the reactions of electrons and defect electrons are bimolecular as a result of dissociation and recombination – that is: being nonlinear. Including also the reactions of higher than linear order, the formulation by means of matrices does not suffice; then, a comprehensive analytical representation of the reaction system is achieved by means of tensors.
    The expansion of the matrix-based analytical description of the photographic micro-process to tensors stands for more precise consideration of different influences but does not mean another quality. The exposure-matrix goes over to an exposure-tensor and the multiplication of matrices will be carried out by transition tensors – always non-commutatively.
    The analytic representation of the photographic process by reaction tensors is used as a prototype for dynamic processes with all manner of interconnecting relations and reaction orders. Matrices are defined as special tensors of second order. Sheer matrix algebra can be used advantageously in most cases for its mathematical convenience, if the reaction systems are suited to be linearized approximately as being given for short time intervals and continued matrix multiplication.

    Generalization: The Photographic Process
    is a prototype for causally consecutive processes!
    Statements and Conclusions:
    •   Causally consecutve processes occur in reaction systems with functional relations among the compartments and influences from inside and outside.
    •   A typical phenomenon of the photographic process is Schwarzschild's blackening law, which is caused by the consecutive endoenergetic processes acting during the exposure of photo-sensitive material to light.
         However, this well-known law is not confined solely to photography; it is, moreover, a general physicochemical law with forward and backward reactions in a multistep reaction chain, which come partly in equilibrium states by reducing the order and prolonging or shortening the duration of the reaction time.
         The generalization of Schwarzschild's law includes furthermore exoenergetic processes, which are supported by the material and energetic content of the reaction system.
    •   In the course of the reaction process the compartments of the system will be taken over from the past, newly created and redistributed – accumulating the results.
         The compartments of the system can be coordinated to the axes of a hypergeometric space in form of a multidimensional vector framework – including the three-dimensional Euclidean space. Changes of the composition are described as turning-streching of the resultant main vector, which is mathematically performed as a vector transformation.
         By outer and/or inner energetic influences the composition of the compartments and their mutual relations run through a series of qualitatively different stages in causal consecution, which is defined as a composition transformation.
    •   The generalized photographic process does not require any reaction chain, for it holds also for networks with branching and separation by inner and outer regions of conjuncted compounds with differently acting influences.
    •   The preconditions of processes are determined by a static network of spatial, structural, and functional connections with the potential reserves of matter and energy. The processes move dynamically on specified paths. The course of the process creates new states, structures, and material compositions, which again form the basis for subsequent processes. Static and dynamic conditions can exist at the same time and influence each other.
    •   The transition of the compound system in a sequence from one stage (step) to a following one is performed – schematically, analytically, and numerically – by tensor transformation.
    •   The reaction tensor is a scheme of the transition coefficients, which are put into functional order and contain all external and internal influences onto the reaction system.
         By means of a mathematical treatment, the reaction tensor is converted to the transformation tensor.
         The tensor is composed by interdependent coefficients and independent subtensors, which determine parallel processes.
         A running process is the statistical mean of a multiplicity of similar parallel processes. The statistical quantity of the reaction components is related to the capacity volume of the reaction stages, which corresponds to the magnitude of a concentration.
    •   The transformation by tensors may embrace an amount of different parallel processes.
         A series of similar events need not be causally linked if they belong to independent (parallel) processes.
    •   Causally consecutive processes occur in a series of transformations by events, steps, or steady conversion. There are also sliding steps with a gradual transition.
    •   The stages of the reaction chain can be self-consistent as in the case of coupled oscillators (pages 74-80). With slidingly coupled oscillators, the oscillation spreads out as a wave. Some examples: one-dimensional – rope waves, two-dimensional – water waves, three-dimensional – sound waves, electro-magnetic waves. Local energy concentrations can be caused by interference, resonance, and focusing of waves that lead to line breaks, freak waves, or heat and fire.
    •   Impulse-like or longer-lasting influences can act on the running process at any time that change the course of the process and determine the result once they occur.
    •   The beginning of a change in the course of the process forms a turning point >< , which divides the entire process into two sections – a pre-process and a post-process with different functional relations, transitional coefficients, and compositions. Expressed by the coordinated transition tensors there is:
        T1 >< T2
      Swapping the temporal order of two different onsecutive process sections leads to unequal results.
    •   The chronological reversal of functionally connected process sections is called:
        Commutativity. Following the rules of tensor multiplication in a multidimensional vector space (Anhang, pages 437-438), the exchange of process periods with different transition parameters is generally non-commutative.
      As the tensors of the transition from one process section T1 to the next one T2 is performed by tensor multiplication, there yields the commutativity-relation (page 7, equ. 15):
        T1 . T2 # T2 . T1
    •   Inversion of the sign of the coefficient tensor causes an even functional reflection at the time axis and inverts the transformation. The transformation is performed by inversion of the transition tensor to its reciprocal tensor:
         T –1 = –T
      Thus, with negative time, a sequence of transformations can be traced back from its end to its beginning. Just as the reflection at a mirror, the past appears as a virtual reverse course of events.
          A time reflection (Zeitspiegelung, page 9, equ. 26-31) is a retrospective but no causal reality – because:
          Time is a one-way street.
          In the course of real processes in a closed reaction system, full time reflection does not take place because of loss of information due to interaction of the compartments varied during the process.
    •   Time reflection can be very useful in determining the origin of a process and looking for the cause of things like disease or poisoning – provided that the transition conditions are known. – (Application examples: medicine, diagnosis, pathology, criminology, forensis, etc.).
    •   The tracing of a process from a period of time is also for science and research of great interest.
      A famous example is the back calculation of the currently still observed expansion of the Universe (Hubble 1927) down to the origin from one point – the creation out of nothing – by Lemaitre (1925). Our knowledge begins with a “Big Bang” even in many areas.
    •   A generalized process applies also to interrupted, intermittent, double, and multiple processes, the particular ones of which need not to be continuous in time but may even shrink to transformation events.
    •   Many processes take place by rapid changes occurring occasionally in a sequence of causally connected stages by some kind of a schedule of appointments.
    •   Likewise as for continuous processes, also for discrete processes but with consecutive sequences of tensor transformations there are valid the laws of non-commutativity and time reflection or rather causal-sequence reflection.
    •   With transition coefficients given for every single transformation before, forward processing can be calculated in any case; the inversion, however, is usually uncertain. For a correct inversion, all tensor transformations have to be rewound in their reverse sequence with negative time progression – or: time regression.
    •   For the preliminary calculation of processes, the past is certain but the future is uncertain. The process running through is known and can be analyzed by interpolation for its components and reaction velocities. The afterwards connected process is an extrapolation of the conditions at a definite time and the following interaction of different components and influences.
    •   The analytical representation of processes by tensors is limited to tensors of the second order – namely matrices.
      Tensors of third and higher order are filled with coefficients as functions of the components resulted in the process running before. This leads to an incalculable entangling of all ingredients of the system and irreversible mixing with the later evolved coefficients. (The transition from the second to the third order and more is problematic even elsewhere as for example: cubic equation, three-body problem.)
    •   Besides constant conditions there can occur impulse-like and rapid, periodic and intermittent, sporadic and accidental, stochastic and chaotic changes (pages 10-16) of the transition coefficients, forcing their rhythms, shocks, and noise on the course of the process, though being reflected in their results by resonance and retardation response of phases and gravity centers.
    •   Real processes are subjected to different physical influences acting during the reaction time from outside or/and inside the reaction system – as there are:
      1. Influences from outside like temperature, pressure, density, radiation (light, X-rays, alpha-, beta-, gamma-rays) – defining endoenergetic processes.
      2. Influences from inside like heating, explosion, chemical processes, splitting, reproduction, multiplying, spreading, etc. – defining exoenergetic processes.
      3. Influences by unforeseen or deliberately caused accidents like ignition by lightning or arson, outbreak of diseases, pandemics, etc. – defining accidental processes.
      4. Mutual influences of parallel processes by coupling or subordinate dependence like courses of life and organs, periodical seasons of the year and vegetation, catalytic action, etc. – defining coupled processes.
      5. Self-influenced processes coming in being without recognizable origin like self-ignition, diseases, epidemics, emergence and evolution of life, Creation of the Universedefining autogenerating processes.
    •   Comments:
      ° The processes under the points 1. and 2. occur as single or causally connected reactions.
      ° Exoenergetic processes (like the flaring up and spreading of fire) and endoenergetic processes (like the extinguishing of fire) are mostly present together in complex reaction systems.
      ° The physical concept of energy is generally transferred to other process-driving magnitudes. There are initiating "energetic causes" for the development of material as well as immaterial processes like education, economy, politics, etc. In many processes a quantity of diverse things and even individuals works as a source of "universal energy".
      ° The processes under point 3. give the initial set off for an independent consecution on reaction-specific conditions at the expense of the material and energetic substance and the resources of the reaction sytem.
      ° A process has a past history, a result, and often there is an aftermath.
      ° The processes under point 4. occur as connected reactions by interaction with stimulating stochastic and/or periodic impulses, leading by feedback coupling to self-consistent oscillations (like the rhythmical heartbeat).
      ° Usually processes are interlocked in a hierarchy of coupled sub-processes which correspond to subtensors within a global tensor.
      ° The reaction results of subordinate processes determine the transition coefficients of interacting superordinate processes.
      ° Back-coupled and cross-coupled reaction systems oscillate by feedback with eigenfrequencies determined by the structure and coupling parameters.
      ° The processes under point 5. start from a state of equilibrium or chaos – provided there exists already a potential reaction system with the suited functional relations and material and energetic resources.
      ° Due to the internal energy, tiny deviations from equilibrium occur, which are caused by nearby system-specific attractors. The fluctuations will be devided repeatedly by bifurcations organizing a new processing order, thereby increasing the reaction system exoenergetically in extent, quantity, and force.
    •   Every process proceeds under the structural, directional, logical, interrelated, promoting, and restraining conditions of the special medium: – the reaction system.
    •   All causally consecutive processes are based on diversely structured reaction systems:
      ° There are isolated reacting isles, embedded in a complex network together with other reaction regions – but normally linked whith input and output from or to outside.
      ° Besides of manifold parallel reaction regions, there are symbiotic and hybrid zones, which work continuously and/or occasionally together.
      ° The real basis of reaction systems is usually material and natural – as for example: the silver-halide crystals in photography – but this is also the case for nature in general, physics, chemistry, biology, geology, etc.
      ° The basis of reaction systems can even be non-material – as for example: history, politics, justice, social system, plans for life and education, music, management of economy, traffic, building, etc.
    •   The classic photographic process is restricted to endoenergetic reactions in the structural medium of crystalline silver halide as the material reaction system.
    •   A process stands for change, development, progression, redistribution, conclusion, and end result of the contained compartments and their interconnecting relations.
    •   The process progresses in time, moving from imbalance to equilibrium. During the progressing the mixing of transitions and components increases up to homogenization unless there appear and act new external influences or there will be ignited exoenergetically new internal secondary processes, maybe with chain reactions (examples: locomotion, extensive fire, interference and focusing of waves – freak waves, explosion, nuclear power, biological reproduction, proliferation, spreading of epidemics and pandemics, freak waves, transport of infection to distant areas).
          The progression of processes is limited by backward transitions which consist in braking, moderation, isolation, slowing down or even stopping (examples: fighting against fire, nuclear reactor moderation, break down of biological reproduction, toxic drugs, medical treatment, immunization, preventive and healing measures against epidemics and pandemics with the best prevention: avoiding contact, keeping distance! ).
          Only for accidental influences there exists neither prevention nor forecast. After having been triggered, the reactions go on by the intrinsic functional relations of the system. The chronological coordination to the consecutive course of the process is a matter of destine.
    •   Application, exploitation, and consequences of processes by human activity are: cooking, agriculture, industrial work, education, climate, instrumentalism (war, pandemics, politics).
    •   The mathematical treatment and the computer-aided calculation of reaction systems by means of tensor algebra becomes more and more complicated the higher the order and the more numerous the compartments are.
    •   Considering the complexity of reaction systems, the practical computation of tensorially formulated processes needs some significant simplifications. – Comments:
      ° The calculation of processes always relates to the entire reaction system.
      ° Even with causally consecutive reactions, the transitions are not calculated successively because they are treated as a unit by tensors and matrices.
      ° The beginning of a functional analysis of a reaction system is the drawing of a process flow diagram. This gives a complete overview (see: article, page 435, figure) about the arrangement and relationships between the components and their stages.
      ° The tensor formulation of the system structure is mathematically expressed by a single transformation from covariant to contravariant coordinates (page 427, equ. 10-13) with computer-compatible algorithms.
      ° During a progressing process a lot of tensorial transformations have to be carried out, using every time a standard algorithm.
      ° An always passable way to solve steadily evolving processes is given by the conversion of the system of differential equations (see page 427, equ. 10) in integral equations (see page 428, equ. 14), expanding them into series, which will be broken off after the second (linear) term (see page 430, equ. 26). The progress of the reaction up to the final result is achieved by repeatedly multiplying the infinitesimal tensor factors.
      ° A particular favorable way of treating a reaction system of second order is given by expanding a series of a matrix exponential (see exposure matrix, page 265, equ. 22-23), since for this there exists a standard algorithm.
      ° The matrix-based reaction system can be used also for processes of higher than second order. Then, in the differential equations the components will be added multplicatively to the transition coefficients. With regard to the required convergence of the series the time intervals should be as short as necessary.
    •   A good way out of the dilemma of increasing mathematical effort with the complexity of the system is resorting to probability calculation, since every transition from a certain condition to another one corresponds statistically in a group of compartments to a probability factor.
      The statistics bundles and averages a multiplicity of parallel consecutive reaction chains, making the even transition numbers of the reaction orders odd.
          The article demonstrates an example of a rather simple system, which can be mathematically calculated but could also be confirmed by probability calculation.
    •   Usually, forward and backward reactions take place simultaneously. A single transition from one stage to the next one fulfills a reciprocity law as the probability product of the generalized driving power P multiplied by the reaction time t
          P . t = const .
    •   If for a sequence of consecutive transitions the balance of build-up and build-down is biased, then the probability product power .time yielding a constant work-effect deviates from reciprocity.
          According to the multiplication rule of consecutive probabilities and with the transition numbers for both magnitudes m and n, yielding a reciprocity criterion by
          m/n <=> 1,
      there results the characteristic product of exponentials:
          Pm . tn = const
      This is the General Schwarzschild-Law !
      For discrete transitions, instead of the time t a term of temporally arranged transformation events is to be inserted.
          In addition to using the probability calculation, the Schwarzschild power product E . tp can be analytically derived from the transition tensor, as shown by the exposure matrix. This is not a practicable way but it proves the correctness of the mathematical treatment.
    •   The Schwarzschild law predicts the outcome of a process on the basis of probability calculation.
      Hereby the information about the structure and the course of the process is covered. That concerns mainly the non-commutativity of double exposures, the time and structure reflection, and coupling to other processes.
      Predicting the exposure of photographic plates – above all the forecast – was the main purpose of studying the properties of photographic materials for use in astrophysical observation and measurement by
      Karl Schwarzschild.
    •   Tensor transformation and probability are not opposites, but two sides of the same coin.
    •   The course of consecutive processes cannot be traced back in any case due to loss of information by statistics and mixing of all participating ingredients and relations of the reaction systems.
    •   The transition process in direction to balance of all interacting forces can physically be described as an increase of entropy, limiting any forecast, so as it is known of the highly nonlinear meteorological processes.
    •   Nevertheless, a process with a more or less approximated forecast is possible to some extent with self-evolving coefficient tensors acting in infinitesimal time intervals and progressing by continued tensor multiplication.
    •   The start of the reaction system from a chaotic state or the transition through an equilibrium is naturally determined by the tiniest atomic movement but mathematically-numerically done by the limiting accuracy of a computer program.
    •   Tensor-determined processes exist functionally in reality with all physical, natural, and logical consequences – independently of the possibility to calculate them mathematically. Even for such processes on the way of degenerating indefinitely into chaos, there are valid the laws of Schwarzschild, non-commutativity, time reflection, periodicity, impulse response, and resonance – quite normally.
    •   In further generalization the Tensor Algebra describes – maybe at least allegorically – various processes in nature, biology, genealogy, geology, meteorology, society, biography, education, jurisdiction proceedings, techniques, economy, and business etc., where continuous processes and/or discrete transformation events are arranged arbitrarily in a causal consecution with oppositely competing forward and backward transitions, which yield the
      Schwarzschild's famous blackening law
      found and explored at photographic plates
      is valid for all kinds of causally consecutive processes!


    P 52. Gerth, E.:
    Zur analytischen Darstellung der Schwaerzungskurve.
    III. Die Reaktionstensoren des photographischen Prozesses

    J. Signal AM 6 (1978) 6, 421-439
    The Reaction Tensors of the Photographic Process

    P 36. Gerth E.:
    Analytische Darstellung der Schwaerzungsfunktion
    mit Hilfe von Matrixfunktionen

    Annalen der Physik 27 (7) (1971) 126-128
    Analytic representation of the photographic
    characteristic blackening curve by matrix functions

    Abstract HTML-Version

    P 41. Gerth E.:
    Reaktionskinetische Prozesse
    der Entstehung des latenten Bildes
    und der photographischen Schwaerzung
    Bild und Ton 26 (1973) 45-48, 59, 69-73, 107-110, 120
    Reaction-kinetic processes
    of the emergence of the latent image
    and the photographic blackening

    The numbers of the references are related to the Register of Published Articles of E. Gerth.

    Last update: 2022, April 17th (17.04.2022)