Doubly-Periodic Atmospheric Power Plants

Doubly-Periodic Atmospheric Power Plants Otto Ziep *Independent Scholar, Berlin, 13089, Germany. Corresponding author(s). E-mail(s): [email protected]; Abstract A cosmic-ray-charge-cloud -superfluid cools atmosphere by a certain amount of newly created matter near zeros of fractal holomorph xiand sigma function. A quadratic model compensates positive energy of molecules mainly by dark, complex phantom energy and negligible ion energy. A proposed doubly-periodic power plant replaces one-periodic cycles in order to gain cooling energy a minimum amount of energy. Created matter is a zero-energy state of biomass and dark energy which is capable to compost into a zero-energy state. Doubly-periodic processed vacuum energy is a non-stationary energy state of a complex Lagrangian lower than that of a physical real field. Keywords: information density universe, doubly-periodic processing, elliptic involution, breathing modes, atmospheric power plant, bifurcation, one-dimensional chaotic map 1 Introduction The present paper describes the technology and the algorithm for an information density universe. Rational coordinates xµand entropy current densities jeare like addresses in a computer memory. The quadratic map deserves attention as a minimal regular map and a maximal regular map. Consecutive γ◦zcreate binary invariants f(ωk) and lattice periods δωkwith complex multiplication (CM) of iterated elliptic curves [1]. Iterated holomorph binary f(ω)=(f1 f2) = dx +idy dz −dt =σµdxµ ds in C∞allow a stereographic holomorphicmeromorphic projection to Riemann sphere 1 with Pauli matrices σmu. A rational mean symplectic f(ω)-subset are Minkowski coordinates where velocity of light c stands for maximally chaotic iterates which are simplest cycles. A complex subset contains one-dimensional γchaotic currents within a fractal zeta universe (FZU). A large number hypothesis (LNH) correlates low and high energies and dimensions [2,3]. As shown in Section 2invariant f(ω) is conformal stress-energy which offers a technological application described in Section 3. The 10∞zoom gauge coupling predicts continuous creation of matter in FZU and lower vacuum densities. The present paper proposes a laboratory to test generation of newly generated matter in FZU which is called doubly-periodic atmospheric power plant (DPAPP). Up to now power plants process subsequent one-periodic cycles. Isothermal and isentropic Carnot processes are e.g. consecutive one-periodic cycles converting vacuum energy ρvac of one-periodic inertial systems in Minkowski spacetime. In quantum statistics ρvac are one-periodic zero-point oscillations ρvac(QS) of periods 11/m where ρvac(QS)≃1050−20ρvac(GR). ρvac(GR) in general relativity is confirmed by experiment. This cosmological constant problem (CCP) is resolvable by unified fields with elliptic involution in an open FZU whereas laws of thermodynamics apply to closed systems [4]. The present paper describes a technology to obtain lower local values ρvac < ρvac(GR) by breathing acting doublyperiodic on a bifurcating potential flow. FZU allows local breathing modes as quasi-stationary vacua to extract heat energy in Section 3. Local creation of matter is analogous to a breathing process of living beings. This local Big Bang-Big Rip scenario would preferentially create molecules H2Oand CO2[4]. Energies comparable to the Planck energy Mpare detectable by e.g. lightning energy of 1GJ ≃10−5g≃0.5Mp≃1028 eV . The present theory of lightning is that opposite charges (positive and negative) accumulate in different regions. At lightning potential breakdown discharge occurs if an accumulating one-periodic electrical potential up to 105−108Vin a thundercloud exceeds the insulating capacity of the air, causing a rapid, massive discharge of electricity known as a lightning flash. This theory is extended to terrestrial gamma ray flashes and cosmic rays (CR) [5]. Recently reported evidence indicates apparent ignition of lightning by cosmic-ray showers or from an avalanche of electrons seeded by extraterrestrial cosmic rays [5]. Analogous to CR-lightning coupling FZU is capable to store dark existing energy as complex dark matter of five unified fields instead of electromagnetic polarization induced by clouds. A FZU cosmic-ray-charged-cloud superfluid (CRCCS) stores unified energy in a nonturbulent open system. Tropical regions with highest lightning rates limit DPAPP processing temperatures to about 50 ° C in order to avoid lightning which would discharge the CRCCSatmospheric capacitor. Every stationary point is surrounded by a three-component doubly-periodic stable state in Section 4. Theory and experiment of the open FZU system is sketched in Section 2and 3. Topological entropy variation by altitude is superimposed by lateral temperature gradients. Already moderate temperature gradients are suspected to induce a bifurcating spacetime (BST) giving complex non-detectable energies [6] [7]. Non-detectable particles are viewed as complex dark vacuum energy. A Planck energy can be stored permanently as a doubly-periodic cycle by avoiding turbulences. DPAPP tests the ability to generate lower vacuum energy 2 density by doubly-periodic processes by ‘creatio ex nihilo’ to a certain amount in CR, atmospheric clouds and photosynthesis in [4]. Cloud motion, plant growth (photosynthesis) and CR are united by regular chaotic iterates of complex BSTcurvature f(ω) near simple zeros znt [4]. Section 6confirms a cloud radiative forcing triggered by negative phantom dark energy. 2 Atmospherean algorithmic chaotic RC circuit FZUvacuum energy is fractal. Low iterated k-components is identified as ultra-high energies above the detection limit which is known as the GZK cutoff. The step k=0 is above the detection limit. Earth ' s atmosphere is viewed as being describable by an entire transcendent, smooth one-dimensional complex function related holomorphicmeromorphic to Riemann sphere in variable z=h+il of altitude h and latitude l in tangential plane. The Weierstrass sigma function σ(z) = −∂jσ(z) ∂z or ξfunction ξ(z) = −∂jξ(z) ∂z of the Riemann zeta function ζ(z) represents a quantum Hall topology (QHT) [8] [9]. QHT dynamical variables are non-analytic potential flow lines ϕσ(x, y), ϕξ(x, y) : ϕσ(x, y) = jσ(z) + ¯ jσ(¯ z) whereas ∂zσ(z) depend on ∂2 zσ(z) and ∂g2,3σ(z) QHT yields the Laplace equation ∂z∂¯zσ(z) = 0. CRCCS is non-turbulent on osculation planes of a twisted cubic Ctw in space. The hyperelliptic Kummer surface K(X(f)) is parametrized by a quadratic Hermite-Tschirnhausen process z←F(t, z)≃γ◦z. Elliptic and hyperelliptic ' current densities ' jσor jξare thought as a flow in an atmospheric spherical capacitor circuit jσ=dZdzσ(z) = d(Q C) = 1 4πεε0 (1 R++ −1 R+− +1 R−+−1 R−− ) (1) with two capacitor stripes znt =±1 2±imnfor nontrivial zeros ξ(znt) = 0 or σ(znt). A quadrupole tensor d(Q C) for complex conjugates ξ(z),¯ ξ(z), ξ(¯ z),¯ ξ(¯ z) depends on four radii R±± creating a moment of inertia. Variable z, the Legendre module λand the Weber invariant f(ω) are interrelated. K(X(f)) points are hyperelliptic ℘-functions parametrized by f=f(ω) X(f) = (1,−f, f2,1) = (℘±±,1) ≃ {jσ(u±, u±},1) ≃(T++, T+−, T−−,1) (2) Like ξ(z) the sigma function is entire and allows a Cayley quotient. σ(u±) = −∂jσ(u±) ∂u±→ −∆jσ(u±) ∆u± The discrete ℘-function in (2) transmits to a Schwarzian derivative {jσ(u±), u±}as conformal stress-energy T±± where {F(t, z), z}= 6∂z∂z′ln∆F ∆z(3) 3 Points on Ctw of K(X) are f-parametrized zeros of σ(u±)σ(v±)) ≃ζ(z)ζ(z′)≃X(f)⊗X(f) = 0 (4) which is quadratic in stress-energy T(u±)[T(z)] which allows quadratic f-iterates for σ(u±) in (4). Like a holomorphic-meromorphic transition dxµ(f) steps k+ 1 and k are involute, i.e. f(ωk+1)f(ωk)≃const. If f(ωk) is diffusive, f(ωk+1) is drifting which requires to map ζ(z) zeros to its poles. resζ(z), the residuum of ζ(z) is proportional to the regulator R(K) = logEν. Its exponential map allows to introduce a complex Lagrangian L[ν] for units Eνwith frequency νof a number field K R(K)≃eRdν[f(ω)]logf(ω)≃eRdν[f(ω)]L(ν[f(ω)]) (5) The ideal residuum resζ(z) = 1 corresponds to an infinite set ν. Residua of the Dedekind zeta function ζ(z, K) are used to map σ(u) zeros for optimal regulator values. ξ(z)-zeros znt differ from σ(u) by replacing the mass mnin znt by ω≃√∆ with discriminant ∆ and performing a multiplication over lattice stripes. Eqs. (2) and (4) prove that f(ωk) unifies curvature, time and temperature. Iterates f(ωk+1)← γ◦f(ωk) is chaotic if {F(t, z), z}<0. γ◦f(ω) is related to rational coordinates by holomorphic functions from Riemann sphere and meromorphic functions from complex plane. Iterates k→k+ 1 is entangled and dynamical in FZU [10]. Accordingly, Minkowski spacetime is a mean field where time is quadrupolar, the scale factor (10) and derivatives ∂zis drift-diffusive. This dualism maps diffusion coefficient Dto the velocity of light c D=1=(∆u)2 ω→(∆u)2 ω2+c→(∆u)2 (∆ω)2→c2(6) Stable orbiting laps and Lorentz-invariant are compatible for f(ω)=(f1 f2) γ→γeFµν [γµ,γν]→γeiσz (7) where deteFµν [γµ,γν]= 1 with Dirac matrices γµ, skew Fµν is equivalent to complex ωk+1 →ω2 k+c. Simplest γ-cycles written for the drift-diffusion term ∆f= (γ− 1) ◦fas γ◦γ=γprove that squared invariants are coordinates. A quadratic map contains alternately coordinates. For fluctuating lattice periods ωcoordinates in space are attached to osculating planes and stationary points. Weierstrass zeta function ζ(u, ωk) in ∆fi=Pj=1,..,4cij∆ζ(uj−u0, ωk), i=1,2,3 require four points j= 1,···,4 [11]. Complex time δt →δt2in (5) is related to additive creation of matter t→t2in LNH [3]. Addition on elliptic curves sets lengths δz equivalent to areas δz2modulo a cubic invariant ϕ3(δz). A cubic congruence δz mod ϕ3(δz)≃δz2≃δz−1qualifies a quadratic map as involution. Bifurcating chaotic line segments dlxy ∈z=h+il store 4 thermal energy ≃ϑ2(uk=akωk, ωk) and drift in a global temperature potential VTglobal (z) = Z∇VTcloud dlxy (8) where dlxy are geometric zeta function sequences. In quantum statistics kinematic temperature Tkin is real mean energy Ekin, i.e. Tkin is one-periodic. Complex Ekin is collisional damping and. In CRCCS complex energy is of the order of rest mass, i.e. both energy (time) and temperature are independent doubly-periodic complex quantities of a superfluid flow. Approximating dlxy in (8) by straight lines between znt is multiplying by a charge e. It is claimed that VTglobal (z) governs time-thermal cycles in atmosphere, biosphere and geosphere which can be written as a drift-diffusion potential VTglobal (z) = T+µn[#znt] (9) The number of zeros n[#znt] with Lagrange parameter µapplies to created charge pairs, i.e. CR, organic aerosols, photosynthesis and vegetational air ions [4]. 3 Breathing DPAPP A Carnot process is a thin rectangle of length t→akttraversed by the number of cycles k. In distinction doubly-periodic complex temperature and entropy is ultra-high complex energy for iterated t←γ◦twith deg(t)=22k. An open system CRCCS DPAPP contains already an amount of matter with highest information densities, e.g. plant branching and atmosphericgaseous-liquid-solid slush around the triple point of water (TPW). Instead of one-periodic electromagnetic nonequilibrium artificial photosynthesis a doubly-periodic process is proposed with complex oscillations of temperature and entropy. Already processing by temperature gradients of 101···102 ° Kon fluctuating time-thermal contours should create complex a finite ρdark. A test of this model is that the density of ionized particles changes. DPAPP is based on a complex contour for a superfluid flow potential (8) and (9) without turbulences. Seasons are simulated changing topological entropy δhtby altitude δh. Simultaneous temperature gradients T(z) induced e.g. changing radiation frequency and intensity in varying directions on complex plane are suspected to induce BST and CR. A fluctuating contour δT δS < 0 replaces a Carnot process rectangle. Instead of plates and contacts of the atmospheric capacitor DPAPP a Carnot battery thermal energy storage (TES) and thermal to power (T2P) are proposed to control a cooling by phantom energy ρdark as sketched in Figure 1. 5 Fig. 1 Breathing mode of DPAPP with drift-diffusing doubly-periodic Carnot process. Lower kinetic energy Ekin is in inversion to higher temperature T at ground level On a closed path between atmosphere and ground level a potential difference Va−Vgis generated if a certain amount of matter is created which is triggered by existing matter. The encircled area by branches the vertical z-plane into a tree of chaotic flow lines. Within FZU five interaction participate. DPAPP operates as a statistical zoom by 1020 ≃226comparable to the Avogadro constant. Quadratic forces in ρmin (4),(12),(3) overwhelm a rest mass threshold which favors statistically organic molecules. Season-dependent DPAPP Big Bang -Big Rip processes create molecules (discriminants [4]) consisting of atoms H, C, O, N [4]. This real matter ρmis compensated by complex dark matter ρdark. Created matter is accompanied by complex phantom energy which induces cooling in DPAPP by a product of low count rate jdark and high energy Edark. DPAPP trigger are existing bifurcating flow lines which are realized at lant growth and near TPW which require low temperature gradients of 102 ° K. Accordingly, ±50 ° Karound 0 ° Ccould induce ultrahigh CR energies and ρion even at ground level [6]. 4 Entangled three-component matter The algorithm is easy to understand by non-stationary distorted contour instead of a constant temperature rectangle with modular unit ϑ2(u=aω, ω) for ωkwith rational a= (a1, a2) in universal covering u. CR is understandable by theta constant sources [4]. Complex energies are identified with ρdark where fractional γcreate singularities at each second step. The experimental atmospheric temperature gradient is S-shaped: 6 The rate at which temperature changes with altitude has negative differential branches which confirms cubic f(ω). According to (4) variable f(ω) iterates the hyperelliptic function ℘±± i.e. stress-energy. The underdetermined complex Lagrangian L(ν[f(ω)]) has an infinity of least squares solutions as a branching tree of doubly-periodic ν[f(ω)]. In natural science the doubly-periodic minimum is seasonal plant growth and birth of livings being. A real minimum L(ν[f(ω)]) = 0 yields regulator index R(K) = 1. This corresponds to imaginary quadratic fields of norm f¯ f= 1 which are used in physical field theory. A pure cubic case R(K)≥1 is not preferred. Cyclotomic extensions of yield local minima R(K) yield infinitely many local minima R(K)≪1 above the theoretical lowest achievable value ρvac = 0 [12]. For quasi-stationary cyclotomic states of a complex Lagrangian with lower vacuum energy density Minkowski spacetime is the real hull due to D= 1 for all iterations. The claim that local L(ν[f(ω)]) minima are charge quanta in nontrivial zeros znt of holomorphic functions ξ(znt) = 0 or σ(u±)=0 as solutions of (4) can be tested by measuring a small amount of δρion.σ(u±)=0 or ∆hσ(z) = 0 allows γ◦zor γ◦uand γ◦ξor γ◦σ(u) with hyperbolic Laplacian ∆h. An algebraic map γbranches into the infinity of stable orbiting laps ( ion pairs ) and unstable bifurcating k-components ( CR ) [12]. Odd k-components and even kcomponents yield diffusive f(ωk) and drifting λ(ωk) charges, alternately. Stable laps as oscillations of the Lyapunovexponent λLaround zero yield the Lagrangian L(ν[f(ω)]) ≃ {F, z}=L(F, ˙ F, ¨ F, ... F) in (3),(4) and (5). The discrete γ-invariant derivative already contains the drift-diffusion scale factor R≃zk+1 −zk ˙ F=∂zF=R= 22k Y i=1 ˙ Fi=eλL(10) ρvac(GR) can be clasiief by the Friedmann solution with Lagrangian L=R˙ R2− ϕ3(R) and polynomial VTglobal ≃ϕ3(z=ϕ2k) (11) (11) is cubic whenever the Feigenbaum renormalized iterated degree 2kpolynomial ϕ2kmeets cubic roots. Energies of k-components should follow a 1/22klaw. Eq. (4) is quadratic in ρvac e.g. δmδm´ ≃0 αm2 ++βm+m−+γm2 −= 0 (12) Masses m±itself are f(ω)- quadratic due to 4. Hyperelliptic based coefficients are integer (αβγ) = (1,−136,10) which recover an electron -to-proton mass ratio up to precision of 10−3[13]. Masses m±≃f(ω), f2(ω) are related to the areal density ρres of residua of iterated f(ωk). Optimized values ρres allow to refine (12) and the conjecture about the fine structure constant α−1 f≃2πδF[14]. Up to second order L(ν[f(ω)] has three-components 1, ∂zand ∂z∂z′acting on logE: (i) real matter ρm≃logf(ω) (ii) inert drift-diffusion fields (∂zlogf = 0) with real ions ρion (iii) complex dark matter ρdark in L(F, ˙ F, ¨ F, ... F)≃ {F(t, z), z}as sketched in Figure 2. ρvac ≃ρm+ρion +ρdark ≃0 (13) 7 A square in ρvac in 4is limited by the identity ϑ4 [00] =ϑ4 [01] +ϑ4 [10]. The real density ρm=ρr+ρexc,vdW is a sum of rest mass ρr, excitonic binding energy and quadratic van der Waals interaction ρexc,vdW which is (3)ρm≃T(f)≃∂f∂f′ln∆F ∆f . The scale factor and ∆F ∆f≃K(λ) is a product of theta constants related to quarter periods K(λ) [15]. The differential equation for K(λ) a local wave function Hermite polynomial with frequency νn. Then, ρm=Pn1 2νnare zero-point oscillations. Rest masses ρrare stable iterates of the Big Bang-Big Rip solution for nine discriminants ∆ reducing to a φ3. The inverse Fermion Green ' s function G−1in quantum statistics is related to F(t, z) = γ◦zin [2]. Fig. 2 Real and complex energies within a BST-environment, variable R is the radius of an apparent singularity (charge, universe) variable z is curvature, ρion contains CR [4] 5 Elliptic involution and various vacuum energy minima Iterated complex unified fields describe a superfluid state of an open universe where self-organization in dissipative structures occurs for closed subsystems, e.g. DPAPP. The claim is that doubly-periodic processed unified fields are capable to exhibit local minima of vacuum energy density. The k-iterated Legendre module λ=1 2+¯ ψλmψ m enters FZU as a four-component Dirac current [6] [15] [4] . FZU is an iterated potential flow where vacuum density ρvac[{F, z}, λ] depends on elliptic involution i(λ)=1−λwhich solves the longstanding CCP [12] [10]. Iteration steps γ(ϕ3(f(√∆)) ◦z=F(t, z) with change periods ωvia discriminants ∆ and yield a Gaussian-like diffusion process like measured CR. Quadratic transformed periods (1−1 1 1 )ωobey involution f(ω′) = √2/f(ω) . Equivalent periods (1 0 0 1)ω, (0−1 1 0 )ωobey invariances λ→i(λ),1/λ, 1/i(λ),1−1/λ, −λ/i(λ) which are (1 −λi(λ))3/(λi(λ))2=const (14) Elliptic involution undercuts a given vacuum energy in a fractal DPAPP as a non-local correlation between low and high values of dimensionless energy density. 8 For energy density ρcore ≃101gcm−3at ground level and at ρvac ≃10−31 gcm−3 atmospheric level CR air shower moving with velocity of light are correlated with slow plant grow [4]. The density in FZU is like that of a black hole density which is not a well-defined [2] [15] . This uncertainty in density enables a continuous creation of matter in holomorphic environment near zeros of ξ(z) or σ(z) which is equivalent to QHT. 6 Plant growth, Air ions and Cooling in Atmosphere Measured ion fluxes 2 or 20 −30 ≃cm−3s−1ion-pairs at atmospheric or ground level yield simulated and measured concentrations of about 105cm−3[16,17] which are negligible ρion of about 10−11kW h ·cm−3in (13). At cloud-radiative forcing atmospheric cooling is attributed to cloud cover changes. At CRCCS atmospheric cooling is attributed to phantom energy ρdark which compensates created rest mass ρm. OMG energy 1021 eV at GZK cutoff or the Planck energy Mp≃10−5g≃1028 eV content of a lightning is equivalent to heat or cool 12 gwater or an atmospheric cloud of 5˙ 108g by 1 ° K. Phantom dark matter ρdark is claimed as a thermodynamic cooling effect. Lightning consist of 5 −15 Coulomb which are up to 1030 charge quanta, i.e. #znt in CRCCS. Accordingly, a minimal generated matter of 10−5gis capable to induce a strong energy effect felt as cooling within DPAPP. As a result, one has the relations ρion ≪ρdark,ρmand ρdark ≃ −ρm. Vacuum energies ρvac(GR), ρvac(QS) mainly split into positive and negative parts ρm,ρion,ρdark which would be capable to explain the Dirac Sea. The DPAPP energy gain δρdark is achieved by a certain amount of created organic matter which lowers ρdark. In distinction to the CRCCS gain δρdark present bioenergy extracts energy from the excitonic binding energy δρexc,vdW . In distinction to ion-aerosol clear-sky mechanism of condensation nuclei through cloud brightness and cover the CRCCS-cooling effect in VTglobal (z) is due to complex matter ρdark. Within CRCCS zeros ξ(z) = 0 generate compensating negative phantom energy ρdark and positive large rest mass energy. Composting within ξ(z) = 0 states (4) makes the cooling effect disappear [18] [19] [20]. 7 Conclusions A doubly-periodic processing at photosynthesis would be simultaneous one-periodic cycles of complex energy/time superimposed by one-periodic cycles of complex entropy/temperature. Instead of being an exotic state of matter it is a breathing by growing plants and livings being. Most power plants process consecutively oneperiodic. A fluctuating complex time-thermal contour process is capable to achieve a relative lower vacuum density. Ranges of measured vacuum density vary from general relativistic values ρvac ≃1eV ·cm−3, ρvac(CMB)≃0,26 eV ·cm−3, ρvac(star)≃ 0,3eV ·cm−3which are to at ground level. Theory of the CCP ρvac(QS)≃ 1050−20, ρvac(GR) indicates the validity of elliptic involution. CRCCS is based on complex phantom dark energy ρdark of the order rest energy of a molecule but with opposite sign which would be stored in a DPAPP. DPAPP processing as breathing works on the background of existing branching and ramification in clouds and plant 9