Differential Transfer Relations of Physical Flux Density Between

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1. General Physics Flux Density’s Diﬀerential Transfer Relation

From factual relation experienced by the micro ﬂux increment ∆Q(t˙) in transportation process, where ∆Ql(t˙) denotes the loss amount of physical quantity from ∆Q(t˙),

d∆Q(t˙) d∆Q (t˙) ′ = − l t˙ ∈ [t,t ] (1) dt˙ dt˙

′ ′ t d∆Q(t˙) t d∆Q (t˙) dt˙ = − l dt˙ Zt dt˙ Zt dt˙ or

′ ′ ∆Q(t ) ∆Ql(t ) ′ d∆Q(t˙)= − d∆Ql(t˙) (1 ) Z∆Q(t) Z0 there

′ ′ ∆Q(t ) + ∆Ql(t ) = ∆Q(t) (2)

1 1 ′ to multiply Eq. (2) with ∆t or ∆t′ , then take a limit as ∆t and ∆t → 0 dQ(t′) dQ (t′) dQ(t) dt + l = dt′ dt′ dt dt′ dt or J(t′)+ J (t′)= J(t) (3) l dt′ here J(t′) > 0, J(t) > 0. J(t′) and J(t) are instantaneous flux or rate of physical quantity’s flow corresponding to ′ ′ instant t at observing location and instant t at flux source respectively; Jl(t ) is rate of a total accumulated flux loss at instant t′ at detecting position caused by the possible loss amount of physical quantity during the transfer time t′ − t = T (t)= T (t′); and

′ ′ dQl(t ) ′ t 2 ′ ′ dQ (t ) d Q (t˙) dt dQ (t˙) J (t )= l = l dt˙ = d[ l ] > 0 (3′) l ′ 2 dt Zt dt˙ Z0 dt˙

2. Transfer Relation of Longitudinal Length of the micro ﬂow along moving direction

As a particular physics ﬂux, the length can be either compressed or elongated. So except that the micro length ∆ℓ(t˙) could be increased or lengthened whereas the general micro increment ∆Q(t˙) can only be decreased in general due to ∆Ql(t˙), it has same transfer relations as general physics ﬂux has.

d∆ℓ(t˙) d∆ℓ (t˙) = − c t˙ ∈ [t,t + T (t)] (4) dt˙ dt˙

′ ′ t d∆ℓ(t˙) t d∆ℓ (t˙) dt˙ = − c dt˙ Zt dt˙ Zt dt˙ or

′ ′ ∆ℓ(t ) ∆ℓc(t ) ′ d∆ℓ(t˙)= − d∆ℓc(t˙) (4 ) Z∆ℓ(t) Z0

′ ′ ∆ℓ(t ) + ∆ℓc(t ) = ∆ℓ(t) (5) 3

dℓ(t′) dℓ (t′) dℓ(t) dt + c = dt′ dt′ dt dt′

′ ′ dt v (t )+ v (t )= v (t) (6) io c sis dt′

′ ˙ ˙ ′ t d∆ℓc(t) ˙ Where ∆ℓc(t) denotes the compressed length in transportation process whereas ∆ℓc(t ) = t ˙ dt denotes ′ ′ dt the total accumulated length of compression during whole process T = t − t; vc(t ) is a rate ofR total accumulated ′ ′ compressed length passed through observing point; when ∆ℓc(t˙), ∆ℓc(t ) and vc(t ) takes positive values they reﬂect compression related amount respectively, whereas negative values express the opposites’ of compression or elongation related amount. With a variation of the length, have a change of longitudinal density of the ﬂux quantity and relevant consequence been involved.

ρ(t′) ∆Q(t′)/∆ℓ(t′) ∆Q(t′)∆ℓ(t) = = ρ(t) ∆Q(t)/∆ℓ(t) ∆Q(t)∆ℓ(t′) ′ ∆Q(t) − ∆Ql(t ) vsis(t)dt = ′ ′ ∆Q(t) vio(t )dt ∆Q(t′) ∆ℓ (t′) = 1 − 1+ c (6′) ∆Q(t) ∆ℓ(t′)

This relates to interaction between the ∆Q(t˙) and external system and to energy exchange in the transportation process.

3. Energy Transfer and Exchange Relationvc Respect to the ∆Q(t˙)

From factual relations and deﬁned restricted condition or terms in below

d∆E(t˙) d∆E (t˙) d∆E (t˙) = exc − l t˙ ∈ [t,t + T (t)] (7) dt˙ dt˙ dt˙

′ ′ ′ t d∆E(t˙) t d∆E (t˙) t d∆E (t˙) dt˙ = exc dt˙ − l dt˙ Zt dt˙ Zt dt˙ Zt dt˙ or

′ ′ ∆E(t ) ∆E(t ) d∆E(t˙)= d∆Eexc(t˙) Z∆E(t) Z0 ′ ∆El(t )=∆E(t)−∆Eso(t) ′ − d∆El(t˙) (7 ) Z0 there

′ ′ ′ ∆E(t ) − ∆Eexc(t ) + ∆El(t ) = ∆E(t) (8) and dE(t′) dE (t′) dE (t′) dE(t) dt − exc + l = dt′ dt′ dt′ dt dt′ ′ ′ ′ dt or J (t ) − J (t )+ J (t )= J (t) exc l dt′ ′ ′ ′ dt or P (t ) − J (t )+ J (t )= P (t) (9) exc l dt′ 4

Where ∆E(t) and ∆E(t′) are the corresponding total energy carried by the micro flux increment ∆Q(t) and ∆Q(t′) respectively; ∆Eso(t) is a fractional part of ∆E(t) and it is carried by ∆Q(t˙) from the flux source to detecting section as a contributed part of energy flux source to observational energy flux at detecting position. So exchange energy ∆Eexc(t˙) could have positive values for ∆Q(t˙)’s energy gain or negative values for ∆Q(t˙)’s energy loss. ′ ′ ′ dQ(t ) dQ(t) ′ J (t ),J (t)are energy flux which correspond to J(t )= dt′ and J(t)= dt respectively; here J (t ) > 0, J (t) > 0, and ′ ′ t 2 ′ dE (t ) d E(t˙) J (t )= l = dt˙ (9′) l ′ 2 dt Zt dt˙

′ ′ ′ t 2 > 0 energygainof∆Q(t ) ′ dEexc(t ) d Eexc(t˙) ′ ′′ Jexc(t )= = dt˙ < 0 energylossof∆Q(t ) (9 ) dt′ Z dt˙2  t =0 no energyexchangeof ∆Q(t′) with outside  B. Basic Properties — Simultaneity, Independent Conservation and Dependent on Rate of Signal Transfer Time

1. Simultaneity of the Accompanied Variation of ∆Q(t˙), ∆ℓ(t˙),∆E(t˙), the Three Essential Transfer Relations and Time Function Factor

Any change on the physical quantity in the micro increment, longitudinal length, energy carried by ∆Q(t′) and physical or diﬀerential state inside of the micro ﬂux element occurs simultaneously and is inner related.

2. Independent Conservations of Physical Quantity and Energy

Physical quantity and carried energy of a micro ﬂux increment are independently conserved in their transportation process (Ref. Eqs. 1 and 7) and their diﬀerential transfer relations between time domains of source and observers; From equation (1′)

t+T (t) d∆Q(t˙) t+T (t) d∆Q (t˙) dt˙ = − l dt˙ Zt dt˙ Zt dt˙ there its derivative for t′ d∆Q(t′) d∆Q (t′) d∆Q(t) dT (t′) + l = 1 − dt′ dt′ dt dt′ d∆Q(t′) d∆Q (t′) d∆Q(t) dt or + l = (10) dt′ dt′ dt dt′ then ′ t t 2 d∆Q(t˙) d∆Q (t˙) 2 d∆Q(t˙) + l dt˙ = dt˙ Z ′ dt˙ dt˙ Z dt t1 t1

′ ′ ′ ′ ∆Q(t2) + ∆Ql(t2) − ∆Q(t2) = ∆Q(t1) + ∆Ql(t1) − ∆Q(t1) thus ′ ′ ∆Q(t ) + ∆Ql(t ) − ∆Q(t) ≡ 0 (11) Similarly, from equation (7′)

′ ′ t d∆E(t˙) t d∆E (t˙) dt˙ = exc dt˙ Zt′−T (t′) dt˙ Zt′−T (t′) dt˙ ′ t d∆E (t˙) − l dt˙ Zt′−T (t′) dt˙ 5

′ ′ ′ take its derivative for t , then integration on corresponding [t1,t2] and [t1,t2], there ′ ′ ′ ∆E(t ) − ∆Eexe(t ) + ∆El(t ) − ∆E(t) ≡ 0 (12)

3. Flux’s Dependence on the Time Function Factor — Rate of Signal Transfer Time

Differential transfer relations of general physics flux and energy flux are time functions’ derivative dependent (Ref. Eqs. 3 and 7 ), here (Ref. [1])

dt′ dT (t) t′(t)= t + T (t), =1+ ; T (t)= T (t′) dt dt or dt dT (t′) t(t′)= t′ − T (t′), =1 − dt′ dt′ as the time function is reversible and longitudinal length and relative velocity spread transfer relations ′ ˙ ˙ ∆ℓ(t ) − ∆ℓ(t) = ∆ℓ(t) t′ − ∆ℓ(t) t ′ t = vis(t,t˙ − ∆t) − vis(t,t˙ ) dt˙ (13) Z t there

′ t ′ dt ∂vis(t,˙ ris) dtj vio(t ) ′ − vsis(t)= vis(t,t˙ ) dt˙ (14) dt Zt ∂ris dt or

′ ′ ′ ′ t dℓ(t ) vio(t )dt 1 ∂vis(t,˙ ris) dtj ′ = =1+ vis(t,t˙ ) dt (14 ) dℓ(t) vsis(t)dt vsis(t) Zt ∂ris dt

′ with to multiply equations (13) and (14), then take limitation as ∆tj (t˙) and ∆t → 0

′ d∆ℓ(t˙) d∆ℓ(t˙) t − = ais(t,t˙ − ∆t) − ais(t,t˙ ) dt˙ (15) dt˙ ′ dt˙ Z t t t and

′ ′ ′ t ∂vis(t , ris) ′ dt ∂vis(t, ris) ∂ais(t,˙ ris) dtj vio(t ) − vsis(t)= vis(t,t˙ ) dt (16) ∂ris dt ∂ris Zt ∂ris dt or ˙ ′ d∆ℓ(t) ∂vis(t , ris) ′ |t′ ′ ′ t ˙ ˙ ∆tdt˙ ∂ris vio(t )dt 1 ∂ais(t, ris) dtj (t) ′ = =1+ vis(t,t˙ ) dt˙ (16 ) d∆ℓ(t˙) ∂vis(t, ris) vsis(t)dt asis(t)+ ais(t,t) Zt ∂ris dt |t ∆tdt˙ ∂ris

Where ∆tj (t˙) is time interval during which the ∆ℓ(t˙) pass through a speciﬁc point, that has a relative distance ris with respect to signal source; vsis(t) is initial relative velocity of head of the length with respect to signal source, ′ ′ vio(t ) is a relative velocity of the length’s head with respect to observing point; asis(t) and ais(t ) are corresponding accelerations respectively. 6

C. Major Inference From Eq. (3) and (9)

1. General Physics Flux Transfer Function between Two Time Domains

′ ′ t −T (t ) ′ ′ Q(t )+ Ql(t )= Q(t0)+ J(t˙)dt˙ (17) Zt0 or

′ t t ′ [J(t˙)+ Jl(t˙)]dt˙ = J(t˙)dt˙ (17 ) ′ Zt0 Zt0

2. Energy Flux Transfer Function between The Time Domains

′ t t [J (t˙) − Jexc(t˙)+ Jl(t˙)]dt˙ = J (t˙)dt˙ (18) Z ′ Z t0 t0 or

′ ′ t −T (t ) ′ ′ ′ ′ E(t ) − Eexc(t )+ El(t )= E(t0)+ J (t˙)dt˙ (18 ) Zt0 Both of the transfer functions are ever increasing function.

III. ILLUSTRATION AND DISCUSSION

A. Examples of the Relations’ Use

1. Phase Flux Density and Consequent Energy Flux Density

′ d∆ϕl(t˙) dϕl(t ) For wave phase ﬂux, dt˙ ≡ 0, dt′ ≡ 0, from equation (3) there ′ ′ ′ dT (t ) ω (t )= ω(t)[1 − ] dt′ or ′ ′ ′ dT (t ) ω (t ) 1+ = ω(t) (19) dt′ or

′ t t ′ ω (t˙)dt˙ = ω(t˙)dt˙ (20) Z ′ Z t0 t0 and from equation (17), the phase transfer function

t ′ ′ ϕ(t )= ϕ(t0)+ ω(t˙)dt˙ (20 ) Zt0

′ ′ ′ dϕ(t ) dϕ(t) ω (t )= dt′ is observer’s measuring angular frequency; ω(t)= dt is emitting angular velocity or frequency at phase ﬂux source for the same micro phase increment. 7

′ ′ dEexc(t ) From equation (9) and suppose Jexc(t )= dt′ = 0 in non-absorption media,

′ ′ t ′ dT (t ) dJl(t˙) J (t )= J (t) 1 − ′ − dt˙ dt Zt′−T (t′) dt˙ ′ dT (t ) ′ dJl(t˙) = J (t) 1 − − T (t ) ξ dt′ dt˙

′ ′ ′ ξ ∈ [t − T (t ) ,t ] ′ ′ dT (t ) ′ dJl(t˙) or P (t )= P (t) 1 − − T (t ) |ξ (21) dt′ dt˙ According to the electromagnetic wave energy ﬂux relation, observer’s measuring or measured optical intensity not only has been determined by optical energy’s spatial diminishing factor related with T (t), but also depend upon time function’s factor which is determined by a rate of signal transfer time. From equation (19), it’s known that time function’s eﬀect must be considered in frequency-shift relevant measures and data analysis.

2. Charged Particles Flux and Energy Flux in Linac

For charged particles ﬂux in linear accelerator there exists the particles’ number relation of a micro increment

′ ′ ′ ∆N(t ) + ∆Nl(t ) = ∆N(t) and ∆N(t ) 6 ∆N(t) (22) consequently

′ ′ ′ ′ ∆N(t )q + ∆Nl(t )q = ∆N(t)q ∆N(t )q 6 ∆N(t)q (22 )

′ ′ ′ ′′ ∆N(t )mq + ∆Nl(t )mq = ∆N(t)mq ∆N(t )mq 6 ∆N(t)mq (22 )

This means that the number of particles in the micro element will diminish as they are accelerated. However, according to the number’s ﬂux diﬀerential transfer relation

dN(t′) dN(t) dT (t′) dN (t′) = 1 − − l (23) dt′ dt dt′ dt′ at observing location instantaneous number flux can exceed the emitting number flux due to time function related ′ dT (t ) factor when dt′ < 0. Furthermore, as to rf accelerator T (t) or T (t′) is periodic function with a designated τ ∗, T (t + τ ∗) = T (t) and T (t′ + τ ∗)= T (t′); hence on different time domains along higher energy of the micro flux increment, there will be ever increasing high frequency fractional factor in a Fourier analysis on number flux, charge flux, mass flux and energy flux, although number of the particles, and corresponding mass and charge of the specific micro increment will decrease continuously in the transportation process. According to equation (8) the carried energy of the micro increment, ∆E(t˙), can be progressively increased by positive ∆Eexc(t˙) values; then

′ ′ ′ ∆E(t ) = ∆E(t) + ∆Eexc(t ) − ∆El(t ) > ∆E(t) (24)

Compare equation (22′′)

′ ′ ∆M(t ) = ∆M(t) − ∆Ml(t ) 6 ∆M(t) with equation (24), we can make an inference that the energy gain ∆E(t′) − ∆E(t) can not be stored as or converted into any mass of the micro element for ∆M(t′)−∆M(t) 6 0. Then where has been the energy gain stored in the micro ﬂux increment? We will ﬁnd it stored as internal electrical potential energy in the micro increment, and partially converted to the mass center’s kinetic energy of the micro increment [2]. 8

B. Potential Application

A. By means of phase transfer function (Ref. Eqs. 19, 20), the phase function on observer time domain at measuring location of wave interference region can be transformed to phase source time domain on which the angular frequency or velocity is known. Then phase difference on observer time domain is expressed by term of time function and known or designated parameter of phase source [3]. B. Energy flux analysis on frequency-shift astronomical measurement [4]. C. As an analytic approach one longitudinal properties, energy exchange and flux of charged particles in acceleration [2].

IV. CONCLUSION

The differential transfer relations of general physics flux between time domains of source and observer, accompanied with a longitudinal length’s transfer relation, and corresponding energy flux occur or exist simultaneously; and they reflect different aspect of a specified flux increment’s transportation process between source and observer. Time function factor could significantly affect the values of corresponding fluxes on two time domains and itself or rate of it is a necessary fraction of the transfer relations. The differential transfer relations are set up on conversation of various physics quantities independently and shared time transfer relation. Based on the differential transfer relation of flux, its integral form — physical quantity transfer function between the time domains is inferred.

[1] J. Luo, Time function of signal transportation between time domains of signal source and observer, (In Preparation). [2] J. Luo and C. Zhang, Energy differential structure and exchange of specified micro flux increment of charged particles in longitudinal acceleration. [3] J. Luo, Phase difference function and unified equations of steady and non-steady interference. [4] J. Luo, Distance related to red shift caused by dispersion medium and characterastic optical source, (In Preparation).