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

The

articledescribes 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 thearticlereveals a certain universality.

The photographic process− using matrix algebra:

Thematrix formulation of the photographic processgives a reasonable explanation for the fact that double exposures of equal light quantity(E^{.}tintensity,Etime) yield different results if the sequence of short and of long-term partial exposures is exchanged. Thesetare easily formulated by thedouble-exposure-effects, which wasnon-commutative multiplication of matricesat thedefendedTechnical University Dresdenon April 26^{th}, 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. Thegoes over to anexposure-matrixand the multiplication of matrices will be carried out by tensors − alwaysexposure-tensornon-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 consecutiveacting during the exposure of photo-sensitive material to light.endoenergetic processes

However, this well-known law is not confined solely to photography; it is, moreover, awith forward and backward reactions in ageneral physicochemical law, which come partly in equilibrium states by reducing the order and prolonging or shortening the duration of the reaction time.“multistep reaction chain”

Theincludes furthermoregeneralization of Schwarzschild's law, which are supported by the material and energetic content of the reaction system.exoenergetic processes- 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
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.generalized photographic process

- The
in a sequence from one stage (step) to a following one is performed − schematically, analytically, and numerically − bytransition of the compound system.tensor transformation

- The
is a scheme of the transition coefficients, which are put into functional order and contain all external and internal influences onto the reaction system.reaction tensor

By means of a mathematical treatment, theis converted to thereaction tensor.transformation tensor

of the coefficient tensor causes an even functional reflection at the time axis and inverts the transformation. Thus, withInversion of the sign, 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 virtualnegative time.reverse course of events

Such a(time reflectionZeitspiegelung, page 9, equ. 26-31) is a retrospective but no causal reality − because:

.Time is a one-way street

In thein a closed reaction system, full time reflection does not take place because ofcourse of real processesdue to interaction of the compartments varied during the process.loss of information

- Following the rules of tensor multiplication in a multidimensional vector space (
Anhang, pages 437-438), thewith different transition parameters is generallyexchange of process periods.non-commutative- A
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“generalized process”.“transformation events”

- Many processes take place by
occurring occasionally in a sequence of causally connected stages by some kind of arapid changes.“schedule of appointments”

- Likewise as for continuous processes, also for discrete processes but with consecutive
there are valid the laws ofsequences of tensor transformationsandnon-commutativityor rathertime reflection.causal-sequence reflection

- With
given fortransition coefficientseverysingle transformation before, forward processing can be calculated in any case; the inversion, however, is usually uncertain. For a correct inversion,alltensor transformations have to be rewound in their reverse sequence with negative time progression − or:.time regression

- For the
, the past is certain but the future is uncertain. The process running through is known and can be analyzed bypreliminary calculation of processesfor its components and reaction velocities. The afterwards connected process is aninterpolationof the conditions at a definite time and the following interaction of different components and influences.extrapolation

- The
by tensors is limited to tensors of the second order − namelyanalytical representation of processesmatrices.

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 andwith the later evolved coefficients. (The transition from the second to the third order and more is problematic even elsewhere as for example:irreversible mixingcubic equation, three-body problem.)

- Besides constant conditions there can occur
(pages 10-16) of the transition coefficients, forcing their rhythms on the course of the process, though being reflected in their results byimpulse-like, periodic, intermittent and stochastic changesandresonanceof phases and gravity centers.retardation response

- Real processes are subjected to
acting during the reaction time from outside or/and inside the reaction system − as there are:different physical influences

1. Influences from outside liketemperature, pressure, density, radiation(light, X-rays, alpha-, beta-, gamma-rays) − defining.endoenergetic processes

2. Influences from inside likeheating, explosion, chemical processes, splitting, reproduction, multiplying, spreading, etc.− defining.exoenergetic processes

3. Influences by unforeseen or deliberately caused accidents likeignition 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
is restricted tophotographic process.endoenergetic reactions- A process stands for change, development, progression, redistribution, conclusion, and end result of the contained compartments and their interconnecting relations.

- The process
, movingprogresses in time. During the progressing thefrom imbalance to equilibriumof transitions and components increases up to homogenization unless there appear and act new external influences or there will be ignitedmixingnew internal secondary processes, maybe withexoenergetically(examples:chain reactionsextensive 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 amatter of destine.

- The
and themathematical treatmentof reaction systems by means of tensor algebra becomes more and more complicated the higher the order and the more numerous the compartments are.computer-aided calculation

- A good way out of this dilemma is resorting to
, because every transition from a certain condition to another one corresponds statistically in a group of compartments to a probability factor.probability calculation

Thearticledemonstrates 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
as the probability product of thereciprocity lawmultiplied by thegeneralized driving power Preaction 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
yielding apower^{.}timedeviates from reciprocity.constant work-effect

According to the multiplication rule of consecutive probabilities and with the transition numbers for both magnitudesmandn, yielding a reciprocity criterion by

m/n <=> 1,

there results the characteristic product of exponentials:

P^{m}^{.}t^{n}= const

This is theGeneral Schwarzschild-Law !

- The transition process in direction to balance of all interacting forces can physically be described as an
, limiting any forecast, so as it is known of the highly nonlinearincrease of entropymeteorological processes.

- Nevertheless, a process with a more or less
to some extent with self-evolving coefficient tensors acting in infinitesimal time intervals and progressing by continued tensor multiplication.approximated forecast is possible

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 ofTensor-determined processes exist functionally in realitySchwarzschild, non-commutativity, time reflection, periodicity, impulse response, and resonance− quite normally.

- In further
thegeneralizationdescribes − maybe at least allegorically − various processes inTensor Algebranature, biology, genealogy, geology, meteorology, society, biography, education, jurisdiction proceedings, techniques, economy, and business etc., whereand/or discretecontinuous processesare arranged arbitrarily in atransformation eventswith oppositely competingcausal consecution, which yield theforward and backward transitions

.Schwarzschild-effect

Conclusion:

Schwarzschild's famousblackening law

found and explored at photographic plates

is valid for all kinds of causally consecutive processes!

References

P 52. Gerth, E.:

Zur analytischen Darstellung der Schwaerzungskurve.

III. Die Reaktionstensoren des photographischen Prozesses

J. Signal AM 6 (1978) 6, 421-439

Abstract: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

Abstract:Analytic representation of the photographic

characteristic blackening curve by matrix functions

Abstract HTML-Version

P 41. Gerth E.:

Reaktionskinetische ProzesseBild und Ton 26 (1973) 45-48, 59, 69-73, 107-110, 120

der Entstehung des latenten Bildes

und der photographischen Schwaerzung

Abstract: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 Articlesof E. Gerth.

Last update: 2020, February 20

^{th}(20.02.2020)