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Entropy2016,18, 254
Werecognize thebalanceofenergy, linearmomentumandmass.
Remark4. TheHessianmatrix I of−z, consideredas functionofW throughZ, ispositivedefinite [3]. It is
Fishermetric of the InformationGeometry. For the expression (48), it is easy toverify it:
−δMδZ= 1
β (
eint(δβ)2+m ‖ δw− δβm p‖ 2 )
>0 ,
for anynonvanishingδZtaking into accountβ> 0,eint> 0andm> 0. On this basis,we canconstruct a
thermodynamic lengthof apath t →X(t) [21]:
L= ∫ t1
t0 √
(δW(t))TI(t)δW(t)dt ,
whereδW(t) is theperturbationof the temperaturevector, tangent to the space-timeatX(t).Wecanalsodefine
arelatedquantity, Jensen–Shannondivergenceof thepath:
J=(t1− t0) ∫ t1
t0 (δW(t))TI(t)δW(t)dt .
8.Conclusions
Theaboveapproach isnot limited to classicalmechanics but canbeusedasguiding ideas to
tackle therelativisticmechanics. Beyondthestrictapplicationtophysics, it canbe takenassourceof
inspiration tobroachother topicssuchas thescienceof informationfromtheviewpointofdifferential
geometryandLiegroups.Wehopetohavemodestlycontributedto thisaim.
Conflictsof Interest:Theauthorsdeclarenoconflictof interest.
References
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134
Differential Geometrical Theory of Statistics
- Title
- Differential Geometrical Theory of Statistics
- Authors
- Frédéric Barbaresco
- Frank Nielsen
- Editor
- MDPI
- Location
- Basel
- Date
- 2017
- Language
- English
- License
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Size
- 17.0 x 24.4 cm
- Pages
- 476
- Keywords
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Categories
- Naturwissenschaften Physik