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
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.
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 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:
A typical phenomenon of the photographic process is Schwarzschild's blackening law, which is caused by the consecutive endoenergetic processesacting 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. 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 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.
Inversion of the sign of the coefficient tensor causes an even functional reflection at the time axis and inverts the transformation. 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.
Such 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.
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.   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, periodic, intermittent and stochastic changes (pages 10-16) of the transition coefficients, forcing their rhythms 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.
The processes after 1. and 2. occur as single or causally connected reactions.
The processes after 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.   The classic photographic process is restricted to endoenergetic reactions.   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: extensive fire, explosion, nuclear power, biological reproduction, proliferation, spreading of epidemics and pandemics, 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, preventive and healing measures against epidemics and pandemics with the best prevention: 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.
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.
A good way out of this dilemma is resorting to probability calculation, because every transition from a certain condition to another one corresponds statistically in a group of compartments to a probability factor.
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 !
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.
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
P 41. Gerth E.:
der Entstehung des latenten Bildes
und der photographischen Schwaerzung
Bild und Ton 26 (1973) 45-48, 59, 69-73, 107-110, 120
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: 2020, February 20th (20.02.2020)