- Email: [email protected]
Coarse-graining of fluctuations and preservation of normality
/~7~'~'~
:.r~ vT~c~'~:,~y'/~:z~.7Y, Vol. I, pp. 241 to 246. Pergamon Press 1977. Printed in Great Britain.
COARSE-GRAINING OF FLUCTUATIONS AND PRESERVATION OF NORMALITY A Review of At-Power Reactor Noise Theory from the Statistical Physical Viewpoint
Keiichi Saito Department of Atomic Engineering, Faculty of Engineering Hokkaido University, Sapporo, Japan
A well-known ansatz of an established field of statistical mechanics says (ref. i) that the thermal equilibrium state accepts the principle of detailed balance and fluctuations around the maximum entropy obey the Gaussian law fully characterized by a sole parameter -- the variance o. The other sound ansatz is, however, necessary for analyzing such a non-equilibrium noise as is observed in our nuclear reactors. Kubo et al. (2) has asserted that the fluctuations of Markoffian property still preserve Gaussianity at the non-equillbrium state, particularly they remain at the linear stable state. The assertion is called the propagation of the extensive property of the distribution of a macrovariable. The method of system sizeexpansion is derived from the master (Chapman-Kolmogorov) equation. Evolution equation which determine the probable path and the variance around it are successfully applied to analyze various nonlinear, non-equilibrium statistical phenomena (ref. 2 & 3). Based on the Kubo's ansatz, Tomitas (4) have proposed another ansatz that the stationary state accepts the principle of cyclic balance and introduced a new parameter ~, which characterizes the speed or angular momentum of irreversible circulation of fluctuations. Kishida (5) has investigated the physical and practical significance of the a in relation to nuclear reactor control. While linear Markoffian Gaussian processes are described by a set of linear Langevin equations with Gaussian white noise source, there does not always exist such an ideal situation. For example, neutron branching processes sometimes exhibit non-Gaussian counting statistics, a probability density distribution of temperature fluctuations is observed which is not Gaussian, mechanical stops working at core component vibrations exert non-llnear effect and their response to maybe a normal disturbance turns out non-Gaussian, et cetera (ref. 6). Application of autoregressive method (ref. 7) for statistical analysis of dynamic system to our nuclear power plant control (ref. 8) is naturally based on the non-Markoffian description of reactor dynamics. There are three major approach in the discipline of the statistical mechanics. They are the deductive, the stochastic and the intuitive/empirical approach. Figure I shows a schematic view map which correlates the basic reactor noise theory to the statistical mechanics. The deductive approach starts from the Liouville equation, which describes the Markoffian evolution of the microscopic world of atoms and molecules. Coarse-graining of their fine dynamical behaviors of enormously large number of degrees of freedom leads to the macroscopic world of state variables, with which we can manage to get along. Coarse-grainlng is nothing but operation of projection. There are, however, various kinds of projection. In one occation Markoffianity still propagates into our macro-world. Sometimes the variable which has been apparently vanished in a course of operation exerts any effect on the selected macro-variables world. The "hidden" variable possesses memory of the past behaviors of the macro-world and constitutes a non-Markoffian world. Corresponding Langevin equation has generally non-white or colored noise source, although the causality condition is dexterously made preserved by Nishigorl et al. (i0). Morishima (ii) is now developing a new theory of reactor noise which intends to estimate and classify the colorfulness of noise source affected by the hidden or non-observable state variables. The mathematical tool used in the classification is the method of spectral decomposition or eigen-vector representation of a matrix, which appears as a reactor transfer function matrix. The mathematical method has been also successfully applied by Sako (12) to examine moment stabilities and structures of noise patterns.
242
Keiichi Saito
Hidden state variables diminish Markoffianity and whiteness of random source, they give rise to fluctuations in structual parameters as well. The multiplicative noise is not necessarily Gaussian white. The effect of the random parametric excitation on our reactor stability has been recently examined (ref. 13), any further consideration will be necessary to clarify how the hidden state variables amplify any statics of power plant parameters as well as how to identify their statistical property from the basic theoretical point of view.
REFERENCES I. e.g., Landau, L. and Lifshiz, E. (1951) Statistical Physics 2. R. Kubo et al., Fluctuation and relaxation of macrovariables, J. Star. Phys. 9, 51 (1973) 3. S. Kabashima et al., Fluctuation in transient process of electrical oscillation, J. Ph~,s. Soc. Japan 39, 1183 (1975) 4. K. Tomita and H. Tomita, Irreversible circulation of fluctuation, Proc. Theor. Phys. 51 1731 (1974) 5. K. Kishida, Analysis of irreversible circulation in reactor noise, presented at the SMORH-2 (1977) 6. Pacilio, N. et al. eds. (1975) Reactor Noise--Proceedings of the SMORN-I pages 39, 83, 93, 167. 7. R. Aksike, A new look at the statistical model identification, IEEE trans. AC-19, 716 (1974) 8. K. Fukunishi, Diagnostic analysis of a nuclear power plant using multivariate autoregressire processes, Nucl. Sci. En~. 62, 215 (1977) 9. S. Nordholm and H. Zwanzig, A systematic derivation of exact generalized Brownian motion theory, J. Star. Phys. 13, 347 (1975) 10. T. Nishigori et al., Hidden state-variables and a non-Markoffian formulation of reactor noise, J. Nucl. Sci. Technol. 13, 708 (1976) ii. N. Morishima, Observables and non-observables in reactor noise theory, Trans. 1976 annual Meetin~ of the Atomic Energy Society of Japan (in Japanese) D39 12. O. Sako, Fluctuations in a quasi-linear Markovian systems, Proc. Faculty of En~ineerin~ Tokai University 3, 1 (1976) 13. cf. Williams, M. M. R. (1974) Random Processes in Nuclear Reactors, Pergamon Press Akcasu, Z. A. and Karasulu, M., Non-linear response of point-reactor to stochastic inputs, Ann. Nucl. E n e r ~ 3, 11 (1976) Ka1~meshu and Bansal, N. K., Calculation of moments for point reactor without delayed neutrons, J. Nucl. Sci. Technol. 14, 69 (1977) 14. K. Saito, Noise-equivalent source in nuclear reactors, Nucl. Sci. En~n~, 28, 38h (1967) 15. Jan, B. Dragt, Reactor noise analysis by means of polarity correlation, Nukleonik~ 8, 225 (1966) 16. T. Kawai, Interpretation of Flux Shape Transient with Concepts of "Region of Unity" and "Region of Influence", J. Nucl. Sci. Technol., 12, 35 (1975)
CoarselGra~nin~
of
Fluctuations
and
o~'
Preservation
Normality
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Keiichi Saito
244
Appendix
Reactor physical coarse-graining operation and preservation of normality & whiteness
Nuclear reactors inhere collective motions of neutrons, photons, molecules and atoms. We operate coarse-graining upon the cooperative phenomena to set up their descriptive model, or more generally, the world consisting of the proper number of state variables with the appropriate time and spatial scale. Analyzing the world experimentally and/or theoretically can afford to control the nuclear systems. The concept of coarse-graining originates from the discipline of statistical mechanics. Rooted on the concept, such leading principles emerge as the maximum entropy principle, the central limit theorem, the s or ~ expansion method, the scaling theory etc. The techniques associated wlth these principles are utilized to project the world of micro-variables upon the tractable macro-world. We, reactor physicists and engineers, have been carrying out intentionally or unintentionally coarse-graining procedures upon reactor analysis. In the time domain, we observe the most probable path of evolution of reactor dynamics, fl(t) which is controllable, as well as the uncontrollable fined fluctuations around the probable path, 6f(t). Sometimes the fluctuations are coarse~grained and only the probable behaviors are subjected to our analysis. Reactor diagnostics require, however, operation of the delicate Brownian motion, the dependence of which upon the time and the system size, shape & composition is generally different from their dependence of the probable path. In the spatial domain, the most probable path is subjected to the following steps of coarsegraining and is analyzed at the most appropriate level of these steps: i. Classicalization, which leads to the Boltzmann-type transport equation for neutrons and the thermo-hydraulic equations describing the balance of energy, entropy, momentum & mass. 2. Making discrete continuous coordinate such as space, energy, direction to have a multipoint, multi-energy group model as well as making projection of the multi-dimensional space to set up a fewer dimensional world ( discrete model ). 3. Integration, which coarse-grains and smooths out, for example, o frequent scattering processes of neutrons and describes the random walks as diffusion processes ( diffusion model ) ; o influence of a progeny neutron in a certain spatial cell upon the same or the other cell and introduces coupling parameters with proper lag times ( coupled reactor model ) ; o effect of thermo-hydraulics upon neutron multiplication processes and introduces reactivity feedback coefficients with proper lag times ( feedback model ). h. Determination of an integral parameter, P the static reactivity. On the other hand, the actual nuclear system is subjected to instrumentation and measurement. The obtained data are processed to yield an integral observable, ~ the kinetic eigenvalue of the persisting mode of neutrons. Hereupon, coarse-graining procedures are operated which are of instrumental-informational engineering. The above example exhibits how we are utilizing the coarse-graining operation in a generalized sense. Problems arising from the utilization are the followings; i. What is the philosophy adopted in any actual coarse-graining operation upon the confronted physical processes? 2. What kind of properties of the original collective motions is preserved through the course of a certain coarse-graining? 3. Do the processes have any new property after the operation? Showing the procedures of answering the above problems, suppose a nuclear system is at the cold, clesm & calm state ( viz., at the zero-power state in the usual terminology ). i. We must understand the well-known pattern of the power spectral density (PSD) of the observed neutron fluctuations, or more focussing, explain the difference between the pattern of the auto-PSD of a single detector output and that of the cross-PSD of two detectors outputs as well as explain the reason why random scattering and capture processes of neutrons give rise to the white component of the PSD and only random branching processes yield the colored noise. 2. We describe the state of a cold-clean-calm core by neutron and precursor concentration. Coarse-graining in the time domain, we have Markoffian processes with white noise sources which originate from such elementary processes as neutron scattering, capture & fission processes, precursor decay processes, external source emission processes, spontaneous fission processes, etc. The noise sources are white both in the time and the space, and consist of the binary and the single. The former comes from the branching processes.
Coarse-Graining of Fluctuations and Preservation of Normality
245
The other elementary processes compose the single part. The time scale of the coarse-graining adopted in the course of the above Markoffian description is the order of relaxation time of nuclear reactions (-10-1Ssec). Since we concern neutron fluctuations with a much longer range of time (-the inverse of the prompt neutron decay constant ( s-I ) ), we have coarse-Erained the evolution and relaxation processes of any nuclear r~actlon. When we coarse-grain the frequent scattering & transport processes and describe the spatial spreading of neutron cloud as the diffusion processes, the whiteness of noise sources in the time domain is still preserved but the whiteness in the spatial domain is missed ( cf. ref. 14 ). 3. It is found that both the binary and the single noise source yield the colored noise of the neutron population. The neutron population noise is passed through a detector of absorption type, e.g., a compensated ionization chamber. Then, the obtained current fluctuations consist of the colored noise originating from the binary noise source in the neutron field as well as the white noise from the single noise source. Neutron population noise due to the binary source is linearly converted upon the instrumentation procedures, while noise due to the single source is suffered from a non-linear conversion. Similarly, on instumentation with two detectors and processing to yield the cross PSD, the noise due to single source is also non-linearly transformed and removed off. A part of characteristics of the original information of neutron population noise is deformed upon the procedures of the instrumental-informational operation. On the other hand, derivation of the well-known formula of the polarity correlation function requires the Gaussian property of stochastic processes. Verification of the normality of reactor noise is performed before the polarity analysis ( ref. 15 ). The normality is missed after the polarity processing, because the polarity correlation function has the form of 2/wsin-IF(~). If the original noise is Gaussian-Markoffian and has the exponential function type of correlation function F(T), polarity fluctuations have the correlation with non-exponential type. A non-cold-clean-calm reactor noise ( viz., at-power reactor noise in the usual terminology ) is known to be compound phenomena. Their noise sources diverge both in the time ( or frequency ) & the space domain in their respective contribution to the mixed phenomena. The spatial divergence is conceptualized to be "local" and "global" noise at the first SMOP~N. Kawai (16) has presented a sophisticated concept of "region of unity" and "region of influence" in the analysis of the usual reactor kinetics. The region of unity is a region where transient flux behaviors can be approximately regarded to be in phase, while the region of influence means a region where the effect of a localized driving force cannot, at least, neglected. To make a quantitative analysis, Kawai has introduced a new measure of coupling strengh, viz., "degree of unity", which is defined via fractional power excursions caused by arty driving mechanism in a core. The concept can be applied to the analysis of our reactor stochastic kinetics, which constitutes another course of coarse-graining. Another course of coarse-gralnlng starts at setting up an aaalytical model of at-power reactor noise. We can afford to treat explicity only a few macro-varlables to describe the state of our nuclear system. The other variables related to the noise phenomena are hidden and implicitly treated. Explicitly treated macro-variables constitute a coarse-Erained world of the whole involved variables. A certain kind of projection is exerted upon the coarse-@s'aining, and some properties of the original whole world are preserved or missed. Suppose any statistical mechanical coarse-graining yields a set of n macro-stochastic variables { Zk ; k = 1,2,..,n }, which are normally distributed and independent of each other. When theEe independent variables are obtained at coarse-graining in the time domain, they constitute white noise processes. If the set is lacking in independence, a lines/" transformation --- congruent transformation of the original variables vector Z gives rise to a new set of Gaussian variables, which have a diagonal variance-covariance matrix. In other words, the new variables become independent of each other. There is no loss of generality in the presupposition of independence. Now a reactor physical coarse-Eraining operation is performed upon the variable vector Z and a new state variable vector X, which has a fewer number ( m
exp {-(Zk-mk)2/2Ok] ~
where m~and o are the mean and the variance of the normal variable Zk, respectively. indicator of ~he reactor physical coarse-graining C is I x (C) and defined as follows: I (C) = 1 ; C(Z) e ( --,x ) = 0 ; c(Z)
~ (
x,-
).
The
246
Kelichl Salto
The o p e r a t i o n C c a n b e c l a s s i f i e d as follows : o The C, w h i c h p r e s e r v e s b o t h t h e n o r m a l i t y a n d t h e i n d e p e n d e n c e ( w h i t e n e s s ) . T h i s means that if C is expressed by a certain linear operator L, the variance-covariance matrix Z is related with the diagonal variance matrix Z of the original variables and g i v e n b y x LZ L~ , w h i c h i s s t i l l d i a g o n a l , z o The C, w h i c h Z p r e s e r v e s n o r m a l i t y b u t m i s s e s i n d e p e n d e n c e . This is the usual case of linear transform. o The C, w h i c h m i s s e s b o t h t h e n o r m a l i t y a n d i n d e p e n d e n c e . This is natural in the nonlinear transform. The d e s i r a b l e a-Alysis.
property
of normality
and whiteness
is hard to preserve
in the reactor
noise
:.r~ vT~c~'~:,~y'/~:z~.7Y, Vol. I, pp. 241 to 246. Pergamon Press 1977. Printed in Great Britain.
COARSE-GRAINING OF FLUCTUATIONS AND PRESERVATION OF NORMALITY A Review of At-Power Reactor Noise Theory from the Statistical Physical Viewpoint
Keiichi Saito Department of Atomic Engineering, Faculty of Engineering Hokkaido University, Sapporo, Japan
A well-known ansatz of an established field of statistical mechanics says (ref. i) that the thermal equilibrium state accepts the principle of detailed balance and fluctuations around the maximum entropy obey the Gaussian law fully characterized by a sole parameter -- the variance o. The other sound ansatz is, however, necessary for analyzing such a non-equilibrium noise as is observed in our nuclear reactors. Kubo et al. (2) has asserted that the fluctuations of Markoffian property still preserve Gaussianity at the non-equillbrium state, particularly they remain at the linear stable state. The assertion is called the propagation of the extensive property of the distribution of a macrovariable. The method of system sizeexpansion is derived from the master (Chapman-Kolmogorov) equation. Evolution equation which determine the probable path and the variance around it are successfully applied to analyze various nonlinear, non-equilibrium statistical phenomena (ref. 2 & 3). Based on the Kubo's ansatz, Tomitas (4) have proposed another ansatz that the stationary state accepts the principle of cyclic balance and introduced a new parameter ~, which characterizes the speed or angular momentum of irreversible circulation of fluctuations. Kishida (5) has investigated the physical and practical significance of the a in relation to nuclear reactor control. While linear Markoffian Gaussian processes are described by a set of linear Langevin equations with Gaussian white noise source, there does not always exist such an ideal situation. For example, neutron branching processes sometimes exhibit non-Gaussian counting statistics, a probability density distribution of temperature fluctuations is observed which is not Gaussian, mechanical stops working at core component vibrations exert non-llnear effect and their response to maybe a normal disturbance turns out non-Gaussian, et cetera (ref. 6). Application of autoregressive method (ref. 7) for statistical analysis of dynamic system to our nuclear power plant control (ref. 8) is naturally based on the non-Markoffian description of reactor dynamics. There are three major approach in the discipline of the statistical mechanics. They are the deductive, the stochastic and the intuitive/empirical approach. Figure I shows a schematic view map which correlates the basic reactor noise theory to the statistical mechanics. The deductive approach starts from the Liouville equation, which describes the Markoffian evolution of the microscopic world of atoms and molecules. Coarse-graining of their fine dynamical behaviors of enormously large number of degrees of freedom leads to the macroscopic world of state variables, with which we can manage to get along. Coarse-grainlng is nothing but operation of projection. There are, however, various kinds of projection. In one occation Markoffianity still propagates into our macro-world. Sometimes the variable which has been apparently vanished in a course of operation exerts any effect on the selected macro-variables world. The "hidden" variable possesses memory of the past behaviors of the macro-world and constitutes a non-Markoffian world. Corresponding Langevin equation has generally non-white or colored noise source, although the causality condition is dexterously made preserved by Nishigorl et al. (i0). Morishima (ii) is now developing a new theory of reactor noise which intends to estimate and classify the colorfulness of noise source affected by the hidden or non-observable state variables. The mathematical tool used in the classification is the method of spectral decomposition or eigen-vector representation of a matrix, which appears as a reactor transfer function matrix. The mathematical method has been also successfully applied by Sako (12) to examine moment stabilities and structures of noise patterns.
242
Keiichi Saito
Hidden state variables diminish Markoffianity and whiteness of random source, they give rise to fluctuations in structual parameters as well. The multiplicative noise is not necessarily Gaussian white. The effect of the random parametric excitation on our reactor stability has been recently examined (ref. 13), any further consideration will be necessary to clarify how the hidden state variables amplify any statics of power plant parameters as well as how to identify their statistical property from the basic theoretical point of view.
REFERENCES I. e.g., Landau, L. and Lifshiz, E. (1951) Statistical Physics 2. R. Kubo et al., Fluctuation and relaxation of macrovariables, J. Star. Phys. 9, 51 (1973) 3. S. Kabashima et al., Fluctuation in transient process of electrical oscillation, J. Ph~,s. Soc. Japan 39, 1183 (1975) 4. K. Tomita and H. Tomita, Irreversible circulation of fluctuation, Proc. Theor. Phys. 51 1731 (1974) 5. K. Kishida, Analysis of irreversible circulation in reactor noise, presented at the SMORH-2 (1977) 6. Pacilio, N. et al. eds. (1975) Reactor Noise--Proceedings of the SMORN-I pages 39, 83, 93, 167. 7. R. Aksike, A new look at the statistical model identification, IEEE trans. AC-19, 716 (1974) 8. K. Fukunishi, Diagnostic analysis of a nuclear power plant using multivariate autoregressire processes, Nucl. Sci. En~. 62, 215 (1977) 9. S. Nordholm and H. Zwanzig, A systematic derivation of exact generalized Brownian motion theory, J. Star. Phys. 13, 347 (1975) 10. T. Nishigori et al., Hidden state-variables and a non-Markoffian formulation of reactor noise, J. Nucl. Sci. Technol. 13, 708 (1976) ii. N. Morishima, Observables and non-observables in reactor noise theory, Trans. 1976 annual Meetin~ of the Atomic Energy Society of Japan (in Japanese) D39 12. O. Sako, Fluctuations in a quasi-linear Markovian systems, Proc. Faculty of En~ineerin~ Tokai University 3, 1 (1976) 13. cf. Williams, M. M. R. (1974) Random Processes in Nuclear Reactors, Pergamon Press Akcasu, Z. A. and Karasulu, M., Non-linear response of point-reactor to stochastic inputs, Ann. Nucl. E n e r ~ 3, 11 (1976) Ka1~meshu and Bansal, N. K., Calculation of moments for point reactor without delayed neutrons, J. Nucl. Sci. Technol. 14, 69 (1977) 14. K. Saito, Noise-equivalent source in nuclear reactors, Nucl. Sci. En~n~, 28, 38h (1967) 15. Jan, B. Dragt, Reactor noise analysis by means of polarity correlation, Nukleonik~ 8, 225 (1966) 16. T. Kawai, Interpretation of Flux Shape Transient with Concepts of "Region of Unity" and "Region of Influence", J. Nucl. Sci. Technol., 12, 35 (1975)
CoarselGra~nin~
of
Fluctuations
and
o~'
Preservation
Normality
243 0 e @ m
@ e
~
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0 o l
o
®
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=~
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Keiichi Saito
244
Appendix
Reactor physical coarse-graining operation and preservation of normality & whiteness
Nuclear reactors inhere collective motions of neutrons, photons, molecules and atoms. We operate coarse-graining upon the cooperative phenomena to set up their descriptive model, or more generally, the world consisting of the proper number of state variables with the appropriate time and spatial scale. Analyzing the world experimentally and/or theoretically can afford to control the nuclear systems. The concept of coarse-graining originates from the discipline of statistical mechanics. Rooted on the concept, such leading principles emerge as the maximum entropy principle, the central limit theorem, the s or ~ expansion method, the scaling theory etc. The techniques associated wlth these principles are utilized to project the world of micro-variables upon the tractable macro-world. We, reactor physicists and engineers, have been carrying out intentionally or unintentionally coarse-graining procedures upon reactor analysis. In the time domain, we observe the most probable path of evolution of reactor dynamics, fl(t) which is controllable, as well as the uncontrollable fined fluctuations around the probable path, 6f(t). Sometimes the fluctuations are coarse~grained and only the probable behaviors are subjected to our analysis. Reactor diagnostics require, however, operation of the delicate Brownian motion, the dependence of which upon the time and the system size, shape & composition is generally different from their dependence of the probable path. In the spatial domain, the most probable path is subjected to the following steps of coarsegraining and is analyzed at the most appropriate level of these steps: i. Classicalization, which leads to the Boltzmann-type transport equation for neutrons and the thermo-hydraulic equations describing the balance of energy, entropy, momentum & mass. 2. Making discrete continuous coordinate such as space, energy, direction to have a multipoint, multi-energy group model as well as making projection of the multi-dimensional space to set up a fewer dimensional world ( discrete model ). 3. Integration, which coarse-grains and smooths out, for example, o frequent scattering processes of neutrons and describes the random walks as diffusion processes ( diffusion model ) ; o influence of a progeny neutron in a certain spatial cell upon the same or the other cell and introduces coupling parameters with proper lag times ( coupled reactor model ) ; o effect of thermo-hydraulics upon neutron multiplication processes and introduces reactivity feedback coefficients with proper lag times ( feedback model ). h. Determination of an integral parameter, P the static reactivity. On the other hand, the actual nuclear system is subjected to instrumentation and measurement. The obtained data are processed to yield an integral observable, ~ the kinetic eigenvalue of the persisting mode of neutrons. Hereupon, coarse-graining procedures are operated which are of instrumental-informational engineering. The above example exhibits how we are utilizing the coarse-graining operation in a generalized sense. Problems arising from the utilization are the followings; i. What is the philosophy adopted in any actual coarse-graining operation upon the confronted physical processes? 2. What kind of properties of the original collective motions is preserved through the course of a certain coarse-graining? 3. Do the processes have any new property after the operation? Showing the procedures of answering the above problems, suppose a nuclear system is at the cold, clesm & calm state ( viz., at the zero-power state in the usual terminology ). i. We must understand the well-known pattern of the power spectral density (PSD) of the observed neutron fluctuations, or more focussing, explain the difference between the pattern of the auto-PSD of a single detector output and that of the cross-PSD of two detectors outputs as well as explain the reason why random scattering and capture processes of neutrons give rise to the white component of the PSD and only random branching processes yield the colored noise. 2. We describe the state of a cold-clean-calm core by neutron and precursor concentration. Coarse-graining in the time domain, we have Markoffian processes with white noise sources which originate from such elementary processes as neutron scattering, capture & fission processes, precursor decay processes, external source emission processes, spontaneous fission processes, etc. The noise sources are white both in the time and the space, and consist of the binary and the single. The former comes from the branching processes.
Coarse-Graining of Fluctuations and Preservation of Normality
245
The other elementary processes compose the single part. The time scale of the coarse-graining adopted in the course of the above Markoffian description is the order of relaxation time of nuclear reactions (-10-1Ssec). Since we concern neutron fluctuations with a much longer range of time (-the inverse of the prompt neutron decay constant ( s-I ) ), we have coarse-Erained the evolution and relaxation processes of any nuclear r~actlon. When we coarse-grain the frequent scattering & transport processes and describe the spatial spreading of neutron cloud as the diffusion processes, the whiteness of noise sources in the time domain is still preserved but the whiteness in the spatial domain is missed ( cf. ref. 14 ). 3. It is found that both the binary and the single noise source yield the colored noise of the neutron population. The neutron population noise is passed through a detector of absorption type, e.g., a compensated ionization chamber. Then, the obtained current fluctuations consist of the colored noise originating from the binary noise source in the neutron field as well as the white noise from the single noise source. Neutron population noise due to the binary source is linearly converted upon the instrumentation procedures, while noise due to the single source is suffered from a non-linear conversion. Similarly, on instumentation with two detectors and processing to yield the cross PSD, the noise due to single source is also non-linearly transformed and removed off. A part of characteristics of the original information of neutron population noise is deformed upon the procedures of the instrumental-informational operation. On the other hand, derivation of the well-known formula of the polarity correlation function requires the Gaussian property of stochastic processes. Verification of the normality of reactor noise is performed before the polarity analysis ( ref. 15 ). The normality is missed after the polarity processing, because the polarity correlation function has the form of 2/wsin-IF(~). If the original noise is Gaussian-Markoffian and has the exponential function type of correlation function F(T), polarity fluctuations have the correlation with non-exponential type. A non-cold-clean-calm reactor noise ( viz., at-power reactor noise in the usual terminology ) is known to be compound phenomena. Their noise sources diverge both in the time ( or frequency ) & the space domain in their respective contribution to the mixed phenomena. The spatial divergence is conceptualized to be "local" and "global" noise at the first SMOP~N. Kawai (16) has presented a sophisticated concept of "region of unity" and "region of influence" in the analysis of the usual reactor kinetics. The region of unity is a region where transient flux behaviors can be approximately regarded to be in phase, while the region of influence means a region where the effect of a localized driving force cannot, at least, neglected. To make a quantitative analysis, Kawai has introduced a new measure of coupling strengh, viz., "degree of unity", which is defined via fractional power excursions caused by arty driving mechanism in a core. The concept can be applied to the analysis of our reactor stochastic kinetics, which constitutes another course of coarse-graining. Another course of coarse-gralnlng starts at setting up an aaalytical model of at-power reactor noise. We can afford to treat explicity only a few macro-varlables to describe the state of our nuclear system. The other variables related to the noise phenomena are hidden and implicitly treated. Explicitly treated macro-variables constitute a coarse-Erained world of the whole involved variables. A certain kind of projection is exerted upon the coarse-@s'aining, and some properties of the original whole world are preserved or missed. Suppose any statistical mechanical coarse-graining yields a set of n macro-stochastic variables { Zk ; k = 1,2,..,n }, which are normally distributed and independent of each other. When theEe independent variables are obtained at coarse-graining in the time domain, they constitute white noise processes. If the set is lacking in independence, a lines/" transformation --- congruent transformation of the original variables vector Z gives rise to a new set of Gaussian variables, which have a diagonal variance-covariance matrix. In other words, the new variables become independent of each other. There is no loss of generality in the presupposition of independence. Now a reactor physical coarse-Eraining operation is performed upon the variable vector Z and a new state variable vector X, which has a fewer number ( m
exp {-(Zk-mk)2/2Ok] ~
where m~and o are the mean and the variance of the normal variable Zk, respectively. indicator of ~he reactor physical coarse-graining C is I x (C) and defined as follows: I (C) = 1 ; C(Z) e ( --,x ) = 0 ; c(Z)
~ (
x,-
).
The
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Kelichl Salto
The o p e r a t i o n C c a n b e c l a s s i f i e d as follows : o The C, w h i c h p r e s e r v e s b o t h t h e n o r m a l i t y a n d t h e i n d e p e n d e n c e ( w h i t e n e s s ) . T h i s means that if C is expressed by a certain linear operator L, the variance-covariance matrix Z is related with the diagonal variance matrix Z of the original variables and g i v e n b y x LZ L~ , w h i c h i s s t i l l d i a g o n a l , z o The C, w h i c h Z p r e s e r v e s n o r m a l i t y b u t m i s s e s i n d e p e n d e n c e . This is the usual case of linear transform. o The C, w h i c h m i s s e s b o t h t h e n o r m a l i t y a n d i n d e p e n d e n c e . This is natural in the nonlinear transform. The d e s i r a b l e a-Alysis.
property
of normality
and whiteness
is hard to preserve
in the reactor
noise