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

Thearticledescribes 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, 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

thespread of epidemics into pandemics.Further applications of the generalized process theory: The functional mechanism of epidemics and pandemicsPandemic Freak Waves

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 byexposure-tensor– alwaystransition tensors.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 processesoccur inreaction systemswithfunctional relationsamong the compartments andinfluencesfrom inside and outside.

- A typical phenomenon of the
isphotographic processSchwarzschild'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 – 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
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
preconditions of processesare determined by aof spatial, structural, and functional connections with the potential reserves of matter and energy. The processes movestatic networkondynamicallyThe 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.specified paths.

- 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

The tensor is composed by interdependent coefficients and, which determineindependent subtensors.parallel processes

A running process is the statistical mean of a multiplicity of similar parallel processes. Theof the reaction components is related to the capacity volume of the reaction stages, which corresponds to the magnitude of astatistical quantity.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

occur in a series of transformations byCausally consecutive processes, orevents, steps. There are also sliding steps with asteady conversion.gradual transition

- The stages of the reaction chain can be self-consistent as in the case of
(pages 74-80). With slidingly coupled oscillators, the oscillation spreads out as a wave. Some examples: one-dimensional –coupled oscillatorsrope waves, two-dimensional –water waves, three-dimensional –sound waves, electro-magnetic waves. Local energy concentrations can be caused byinterference, resonance, andfocusingof waves that lead toline breaks, freak waves, orheat and fire.

can act on the running process at any time that change the course of the process andImpulse-like or longer-lasting influencesonce they occur.determine the result

- The beginning of a change in the course of the process forms a turning point
><, which divides the entire process into two sections – aand apre-processwith different functional relations, transitional coefficients, and compositions. Expressed by the coordinated transition tensors there is:post-process

T_{1}><T_{2}

Swapping the temporal order of two different onsecutive process sections leads to unequal results.

- The chronological reversal of functionally connected process sections is called:

. Following the rules of tensor multiplication in a multidimensional vector space (CommutativityAnhang, pages 437-438), thewith different transition parameters is generallyexchange of process periods.non-commutative

As the tensors of the transition from one process sectionTto the next one_{1}Tis performed by tensor multiplication, there yields the_{2}(page 7, equ. 15):commutativity-relation

T#_{1}^{.}T_{2}T_{2}^{.}T_{1}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:Inversion of the sign

T= –_{}^{ –1}T_{}

Thus, with, 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

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

- 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
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 togeneralized 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 coefficientsbefore, forward processing can be calculated in any case; the inversion, however, is usually uncertain. For a correct inversion,everysingle transformationalltensor transformations have to be rewound in theirwith negative time progression – or:reverse sequence.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, shocks, and noise on the course of the process, though being reflected in their results byimpulse-like and rapid, periodic and intermittent, sporadic and accidental, stochastic and chaotic 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

4. Mutual influences of parallel processes by coupling or subordinate dependence likecourses 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 likeself-ignition, diseases, epidemics, emergence and evolution of life, Creation of the Universe –defining.autogenerating processes

- Comments:

_{°}The processes under the points 1. and 2. occur as single or causally connected reactions.

_{°}(likeExoenergetic processesthe flaring up and spreading of fire) and(likeendoenergetic processesthe extinguishing of fire) are mostly present together in complex reaction systems.

_{°}The physical concept ofis generally transferred to otherenergy. There are initiatingprocess-driving magnitudes"energetic causes"for the development of material as well as immaterial processes likeeducation, economy, politics, etc. In many processes aworks as aquantityof diverse things and even individualssource 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 therhythmical heartbeat).

_{°}Usually processes are interlocked in a hierarchy of coupled sub-processes which correspond to subtensors within a global tensor.

_{°}The reaction results ofsubordinate processes determine the transition coefficients of interactingsuperordinate 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 byorganizing a new processing order, thereby increasing the reaction system exoenergetically in extent, quantity, and force.bifurcations

proceeds under the structural, directional, logical, interrelated, promoting, and restraining conditions of the special medium: – theEvery process.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: thesilver-halide crystals in photography– but this is also the case fornature 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
is restricted tophotographic processin the structural medium of crystalline silver halide as the material reaction system.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 reactionslocomotion, 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 amatter of destine.

- Application, exploitation, and consequences of
are:processes by human activitycooking, agriculture, industrial work, education, climate, instrumentalism (war, pandemics, politics).

- 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

- 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 fromtocovariantcoordinates (page 427, equ. 10-13) withcontravariant.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(see page 427, equ. 10) indifferential equations(see page 428, equ. 14),integral equations, 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 theexpanding them into series.infinitesimal tensor factors

_{°}A particular favorable way of treating a reaction system of second order is given by expanding a series of a(seematrix exponentialexposure 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 requiredthe time intervals should be as short as necessary.convergence of the series- A good way out of the dilemma of increasing mathematical effort with the complexity of the system is resorting to
, since every transition from a certain condition to another one corresponds statistically in a group of compartments to a probability factor.probability calculation

The statistics bundles and averages a multiplicity of parallel consecutive reaction chains, making the even transition numbers of the reaction orders odd.

Thearticledemonstrates an example of a rather simple system, which can bebut could also be confirmed bymathematically calculated.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 !

For discrete transitions, instead of the timea term of temporally arranged transformation events is to be inserted.t

In addition to using the probability calculation, thecan be analytically derived from the transition tensor, as shown by theSchwarzschild power product E^{.}t^{p}. This is not a practicable way but it proves the correctness of the mathematical treatment.exposure matrix

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– above all the forecast – was the main purpose of studying the properties of photographic materials for use in astrophysical observation and measurement byexposure of photographic plates

.Karl Schwarzschild

Tensor transformation and probability are not opposites, but two sides of the same coin.

- The
cannot be traced back in any case due tocourse of consecutive processesby statistics and mixing of all participating ingredients and relations of the reaction systems.loss of information

- 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

- 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.

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: 2022, April 17

^{th}(17.04.2022)