TOPOGRAPHY is a branch of mathematics that is an extension ofgeometry.> Topology begins with a consideration of the nature of space,> investigating both its fine structure and its global structure. In> particular, topology captures the notion of proximity without theneed> for a notion of distance. (sounds like child support math)>> So I think if none of the formulas that I have posted befor can be> applied, then maybe i need to look at something more defined of> caldculation processes that notes_ NEEDS A NOTION OF DISTANCE> mathematics consideration of the nature of space- (googled that)> <http://www.google.com/search?hl=en&q=WITH+A+NEEDS+A+NOTION+OF+DISTANCE+\> mathematics+consideration+of+the+nature+of+space&btnG=Search>>> The Nature of Space--I> Theodore de Laguna> The Journal of Philosophy, Vol. 19, No. 15 (Jul. 20, 1922), pp.393-407> doi:10.2307/2939625> This article consists of 15 page(s).> <http://links.jstor.org/sici?sici=0022-362X(19220720)19%3A15%3C393%3ATNO\> S%3E2.0.CO%3B2-Z>>>>> [PDF]> arXiv:math-ph/0008018 v1 7 Aug 2000> <http://arxiv.org/pdf/math-ph/0008018>> File Format: PDF/Adobe Acrobat - View as HTML> <http://64.233.167.104/search?q=cache:QpDJSftIDnMJ:arxiv.org/pdf/math-ph\> /0008018+WITH+A+NEEDS+A+NOTION+OF+DISTANCE+mathematics+consideration+of+\> the+nature+of+space&hl=en&ct=clnk&cd=8&gl=us>> considerations (Sect. 2) is that the method of ME has transformedthe> .... Up to now no notion of distance has been introduced on thespace of> states. ...>
Normally one says that the reason it is difficult to distinguish between two
(5
Page 6)
points in say, the real space we seem to inhabit, is that they happen to be
too close together.
It is very tempting to invert the logic and assert that the
two points A and A + dA must be very close together whenever they happen
to be difficult to distinguish. Thus it is natural to interpret g αβ as a metric
tensor [4]. It is known as the Fisher-Rao metric, or the information metric.
A disadvantage of these heuristic arguments is that they do not make explicit a
crucial property of the Fisher-Rao metric, except for an overall multiplicative
constant this Riemannian metric is unique [5][6].
To summarize: the very act of assigning a probability distribution p(xA) to
each point A in the space of states, automatically provides the space of states
with a metric structure.
The coordinates A are quite arbitrary, they need not be the expected values
〈a α
〉. One can freely switch from one set to another. It is then easy to check
that g β are the components of a tensor, that the distance dℓ
2 is an invariant, a scalar. Incidentally, dℓ 2 is also dimensionless.
There is, however, one special
coordinate system in which the metric takes a form that is particularly simple.
These coordinates are the expected values themselves, A
α = 〈a
α
〉. In these coordinates, g αβ = − ∂ 2 S(A) ∂A α ∂A β (9) with S(A) given in Eq.(5) and the covariance is not manifest.
coordinate system in which the metric takes a form that is particularly simple.
These coordinates are the expected values themselves, A
α = 〈a
α
〉. In these coordinates, g αβ = − ∂ 2 S(A) ∂A α ∂A β (9) with S(A) given in Eq.(5) and the covariance is not manifest.
3 Intrinsic dynamics and time
Our basic dynamical principle is that small changes from one state to another
are possible and do, in fact, happen. We do not explain why they happen but, if
we are given the valuable piece of information that some change will occur, we
can then venture a guess, make a prediction as to what the most likely change
will be.
Before giving mathematical expression to this principle we note that large
changes are assumed to be the cumulative result of many small changes. As
the system moves it follows a continuous trajectory in the space of states. We
almost hesitate to call this self-evident fact an assumption, but as the example
of quantum theory shows, trajectories need not exist.
Thus in order to go from one state to another the system will have to move
through intermediate states; in order to change by a distance 2dℓ the system
must have first changed by a distance dℓ.
Suppose the system was in the state A
α
old
= A
α
and that it changes by a small
amount dℓ to a nearby state. We have to select one new state A
α
new
= A
α
+dA
α
from among those that lie on the surface of an n
A
-dimensional sphere of radius
dℓ centered at A
α
. This is precisely what the ME principle was designed to do
[7], namely, to select a preferred probability distribution from within a specified
given set. The only difference with more conventional applications of the ME
principle is the geometrical nature of the constraint.
6
Page 7
We want to maximize ...
Our basic dynamical principle is that small changes from one state to another
are possible and do, in fact, happen. We do not explain why they happen but, if
we are given the valuable piece of information that some change will occur, we
can then venture a guess, make a prediction as to what the most likely change
will be.
Before giving mathematical expression to this principle we note that large
changes are assumed to be the cumulative result of many small changes. As
the system moves it follows a continuous trajectory in the space of states. We
almost hesitate to call this self-evident fact an assumption, but as the example
of quantum theory shows, trajectories need not exist.
Thus in order to go from one state to another the system will have to move
through intermediate states; in order to change by a distance 2dℓ the system
must have first changed by a distance dℓ.
Suppose the system was in the state A
α
old
= A
α
and that it changes by a small
amount dℓ to a nearby state. We have to select one new state A
α
new
= A
α
+dA
α
from among those that lie on the surface of an n
A
-dimensional sphere of radius
dℓ centered at A
α
. This is precisely what the ME principle was designed to do
[7], namely, to select a preferred probability distribution from within a specified
given set. The only difference with more conventional applications of the ME
principle is the geometrical nature of the constraint.
6
Page 7
We want to maximize ...
....One could argue that these theo-
ries would be phenomenological as opposed to fundamental, that within physics
the search for a fundamental mechanics would still be left open. However, in
previous work we have shown [23] that entropic arguments do account for a
substantial part of the formalism of quantum mechanics, a theory that is pre-
sumably fundamental. Perhaps the fundamental theories of physics are not so
fundamental; they are just consistent, objective ways to manipulate information.
References
[1] E. T. Jaynes, Phys. Rev. 106, 620 and 108, 171 (1957).
[2] J. Skilling, “The Axioms of Maximum Entropy” in Maximum-Entropy and
Bayesian Methods in Science and Engineering, G. J. Erickson and C. R.
Smith (eds.) (Kluwer, Dordrecht, 1988).
[3] R. A. Fisher, Proc. Cambridge Philos. Soc. 122, 700 (1925).
[4] C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).
11
Page 12
[5] S. Amari, Differential-Geometrical Methods in Statistics (Springer-Verlag,
1985).
[6] C. C. Rodrıguez, “The metrics generated by the Kullback number” in Max-
imum Entropy and Bayesian Methods, J. Skilling (ed.) (Kluwer, Dordrecht,
1989).
[7] A. Caticha, “Maximum entropy, fluctuations and priors,” in these proceed-
ings.
[8] C. C. Rodrıguez, “Objective Bayesianism and geometry” in Maximum En-
tropy and Bayesian Methods, P. F. Foug`ere (ed.) (Kluwer, Dordrecht, 1990);
and “Bayesian robustness: a new look from geometry” in Maximum En-
tropy and Bayesian Methods, G. R. Heidbreder (ed.) (Kluwer, Dordrecht,
1996).
[9] F. Weinhold, J. Chem. Phys. 63, 2479 (1975); G. Ruppeiner, Phys. Rev. A
20, 1608 (1979) and 27, 1116 (1983); L. Diosi and B. Lukacs. Phys. Rev.
A 31, 3415 (1985) and Phys. Lett. 112A, 13 (1985).
[10] R. S. Ingarden, Tensor, N. S. 30, 201 (1976); R. S. Ingarden, Y. Sato, K.
Sugawa, and M. Kawaguchi, Tensor, N. S. 33, 347 (1979); R. S. Ingarden,
H. Janyszek, A Kossakovski, and M. Kawaguchi, Tensor, N. S. 37, 106
(1982); R. S. Ingarden, and H. Janyszek, Tensor, N. S. 39, 280 (1982).
[11] R. Balian, Y. Alhassid and H. Reinhardt, Phys. Rep. 131, 1 (1986); R.
Balian, Am J. Phys. 67, 1078 (1999).
[12] R. F. Streater, Statistical Dynamics (Imperial College Press, London,
1995); R. F. Streater, Rep. Math. Phys. 38, 419 (1996); R. F. Streater,
Contemporary Mathematics 203, 117 (1997).
[13] L. Onsager, Phys. Rev. 37, 405 and 38, 2265 (1931).
[14] D. Gabrielli, G. Jona-Lasinio and C. Landim, Phys. Rev. Lett. 77, 1202
(1996); R. F. Streater, Open Sys. & Inf. Dyn. 6, 87 (1999).
[15] H. Grabert, Projection Operator Techniques in Nonequilibrium Statis-
tical Mechanics (Springer, Berlin, 1982); R. Kubo, M. Toda, and N.
Hashitsume, Statistical Physics II, Nonequilibrium Statistical Mechanics,
(Springer, Berlin, 1985); D. Zubarev, V. Morozov and G. Ropke, Statistical
Mechanics of Nonequilibrium Processes (Akademie Verlag, Berlin 1996).
[16] B. Robertson, Phys. Rev. 144, 151 (1966); “Application of maximum en-
tropy to nonequilibrium statistical mechanics,” in The Maximum Entropy
Formalism, ed. by R. D. Levine and M. Tribus (MIT Press, Cambridge,
1978).
[17] E. T. Jaynes, “Where do we stand on maximum entropy?” in The Maxi-
mum Entropy Formalism, ed. by R. D. Levine and M. Tribus (MIT Press,
Cambridge, 1978).
12
Page 13
[18] E. T. Jaynes, “Macroscopic Prediction,” in Complex Systems–Operational
Approaches in Neurobiology, Physics, and Computers, ed. by H. Haken
(Springer, Berlin, 1985).
[19] R. Luzzi and A. R. Vasconcellos, Fortschr. Phys. 38, 887 (1990); A. R. Vas-
concellos, R. Luzzi and L. S. Garcia-Colin, Phys. Rev. A 43, 6622 (1991).
[20] E. T. Jaynes, Am. J. Phys. 33, 391 (1965).
[21] J. L. Lebowitz, Physica A 194, 194 (1993); Phys. Today 46(9), 32 (1993).
[22] See e.g. Sect. 7 of Ref.[11].
[23] Ariel Caticha, “Probability and entropy in quantum theory,” in Maximum
Entropy and Bayesian Methods, ed. by W. von der Linden et al. (Kluwer,
Dordrecht, 1999) (online at http://xxx.lanl.gov/abs/quant-ph/9808023;
“Insufficient reason and entropy in quantum theory,” to appear in Found.
Phys. (2000) (online at http://xxx.lanl.gov/abs/quant-ph/9810074).13
ries would be phenomenological as opposed to fundamental, that within physics
the search for a fundamental mechanics would still be left open. However, in
previous work we have shown [23] that entropic arguments do account for a
substantial part of the formalism of quantum mechanics, a theory that is pre-
sumably fundamental. Perhaps the fundamental theories of physics are not so
fundamental; they are just consistent, objective ways to manipulate information.
References
[1] E. T. Jaynes, Phys. Rev. 106, 620 and 108, 171 (1957).
[2] J. Skilling, “The Axioms of Maximum Entropy” in Maximum-Entropy and
Bayesian Methods in Science and Engineering, G. J. Erickson and C. R.
Smith (eds.) (Kluwer, Dordrecht, 1988).
[3] R. A. Fisher, Proc. Cambridge Philos. Soc. 122, 700 (1925).
[4] C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).
11
Page 12
[5] S. Amari, Differential-Geometrical Methods in Statistics (Springer-Verlag,
1985).
[6] C. C. Rodrıguez, “The metrics generated by the Kullback number” in Max-
imum Entropy and Bayesian Methods, J. Skilling (ed.) (Kluwer, Dordrecht,
1989).
[7] A. Caticha, “Maximum entropy, fluctuations and priors,” in these proceed-
ings.
[8] C. C. Rodrıguez, “Objective Bayesianism and geometry” in Maximum En-
tropy and Bayesian Methods, P. F. Foug`ere (ed.) (Kluwer, Dordrecht, 1990);
and “Bayesian robustness: a new look from geometry” in Maximum En-
tropy and Bayesian Methods, G. R. Heidbreder (ed.) (Kluwer, Dordrecht,
1996).
[9] F. Weinhold, J. Chem. Phys. 63, 2479 (1975); G. Ruppeiner, Phys. Rev. A
20, 1608 (1979) and 27, 1116 (1983); L. Diosi and B. Lukacs. Phys. Rev.
A 31, 3415 (1985) and Phys. Lett. 112A, 13 (1985).
[10] R. S. Ingarden, Tensor, N. S. 30, 201 (1976); R. S. Ingarden, Y. Sato, K.
Sugawa, and M. Kawaguchi, Tensor, N. S. 33, 347 (1979); R. S. Ingarden,
H. Janyszek, A Kossakovski, and M. Kawaguchi, Tensor, N. S. 37, 106
(1982); R. S. Ingarden, and H. Janyszek, Tensor, N. S. 39, 280 (1982).
[11] R. Balian, Y. Alhassid and H. Reinhardt, Phys. Rep. 131, 1 (1986); R.
Balian, Am J. Phys. 67, 1078 (1999).
[12] R. F. Streater, Statistical Dynamics (Imperial College Press, London,
1995); R. F. Streater, Rep. Math. Phys. 38, 419 (1996); R. F. Streater,
Contemporary Mathematics 203, 117 (1997).
[13] L. Onsager, Phys. Rev. 37, 405 and 38, 2265 (1931).
[14] D. Gabrielli, G. Jona-Lasinio and C. Landim, Phys. Rev. Lett. 77, 1202
(1996); R. F. Streater, Open Sys. & Inf. Dyn. 6, 87 (1999).
[15] H. Grabert, Projection Operator Techniques in Nonequilibrium Statis-
tical Mechanics (Springer, Berlin, 1982); R. Kubo, M. Toda, and N.
Hashitsume, Statistical Physics II, Nonequilibrium Statistical Mechanics,
(Springer, Berlin, 1985); D. Zubarev, V. Morozov and G. Ropke, Statistical
Mechanics of Nonequilibrium Processes (Akademie Verlag, Berlin 1996).
[16] B. Robertson, Phys. Rev. 144, 151 (1966); “Application of maximum en-
tropy to nonequilibrium statistical mechanics,” in The Maximum Entropy
Formalism, ed. by R. D. Levine and M. Tribus (MIT Press, Cambridge,
1978).
[17] E. T. Jaynes, “Where do we stand on maximum entropy?” in The Maxi-
mum Entropy Formalism, ed. by R. D. Levine and M. Tribus (MIT Press,
Cambridge, 1978).
12
Page 13
[18] E. T. Jaynes, “Macroscopic Prediction,” in Complex Systems–Operational
Approaches in Neurobiology, Physics, and Computers, ed. by H. Haken
(Springer, Berlin, 1985).
[19] R. Luzzi and A. R. Vasconcellos, Fortschr. Phys. 38, 887 (1990); A. R. Vas-
concellos, R. Luzzi and L. S. Garcia-Colin, Phys. Rev. A 43, 6622 (1991).
[20] E. T. Jaynes, Am. J. Phys. 33, 391 (1965).
[21] J. L. Lebowitz, Physica A 194, 194 (1993); Phys. Today 46(9), 32 (1993).
[22] See e.g. Sect. 7 of Ref.[11].
[23] Ariel Caticha, “Probability and entropy in quantum theory,” in Maximum
Entropy and Bayesian Methods, ed. by W. von der Linden et al. (Kluwer,
Dordrecht, 1999) (online at http://xxx.lanl.gov/abs/quant-ph/9808023;
“Insufficient reason and entropy in quantum theory,” to appear in Found.
Phys. (2000) (online at http://xxx.lanl.gov/abs/quant-ph/9810074).13
1–10
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