Transport coefficients in the early universe

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Transport coe?cients in the early universe Jarkko Ahonen 1Department of Physics,P.O.Box 9,FIN-00014University of Helsinki,Finland February 1,2008Abstract We calculate numerically the electrical conductivity σ,heat conductivity κand shear viscosity ηof the hot plasma present in the early universe for the temper-ature interval 1MeV <～T <～10GeV.We use the Boltzmann collision equation to compute all the scattering matrix elements and regulate them by the thermal masses of the t -and u -channel particles.No leading order approximation is needed because of the numerical integration routines used.

1Introduction

The transport properties of the hot plasma present in the early universe are of great interest.The transport coe?cients play a signi?cant role in the phase transitions of the early universe[1,2,3,4],the creation and development of the primordial magnetic ?elds[5,6]and?nally in the creation of the primordial density perturbations and therefore in the galaxy formation[7,8].

There has been many attempts to estimate the transport coe?cients.(Textbook estimates for thermal conductivity,shear and bulk viscosity in the ultrarelativistic plasma were given in[9]and[10].)In[11]the viscosities of a pure gluon plasma and of a quark-gluon plasma were computed in the weak coupling limit from a variational solution to the Boltzmann equation.In[12]the transport coe?cients were calculated for plasmas interacting through strong,electromagnetic and weak interactions to lead-ing order in the interaction strength.The rates of momentum and thermal relaxation, electrical conductivity and viscosities of quark-gluon and electrodynamic plasmas were also included in[12].In[13]the transport coe?cients and relaxation times were cal-culated to leading orders in the coupling constant for degenerate quark matter within perturbative QCD for temperatures and inverse screening lengths much smaller than the quark chemical potential.The viscosities of the QCD-plasma were considered in [14].Textbook estimates for thermal conductivity and viscosity in the ultrarelativistic plasma were given in[9]and[10].The conductivity of a relativistic plasma has also been considered in[15]by reformulating the collision operator for a relativistic plasma in terms of an expansion in spherical harmonics.

However,all the previous estimates involve approximations.The calculations have been performed only to the leading(logarithmic)orders and certain scattering reac-tions present in the hot primordial plasma have been neglected.In the present paper, the only approximation we make is to assume that when particles appear in the heat bath of the primordial plasma,their thermal velocities are high enough to allow us to treat them ultrarelativistic and therefore massless.We consider radiation dominated plasma with temperatures from about1MeV to about10GeV.

The viscous damping and heat conducting e?ects a?ect the?rst order phase tran-sitions in the early universe.During such phase transitions,instabilities may occur when the transport of latent heat is dominated by the?uid?ow.In[1]this was studied in the EW transition in the small velocity limit,in[2]in the QCD transition,in[3] for cosmological detonation fronts and in[4]for general?rst order transitions with either large or small bubble wall velocities.The instabilities can be damped by?nite viscosity and heat conductivity due to the di?usion of radiation on small length scales.

When considering the creation and development of primordial magnetic?elds it is also of importance to know the transport coe?cients of the surrounding primordial

plasma[5,6].Finite electrical conductivity leads to the di?usion of magnetic?elds and therefore is of importance when trying to explain the further evolution of an initial seed magnetic?eld(see e.g.[16]).The instabilities in the?rst order phase transitions can be shown to create seed magnetic?elds[6].Therefore the seed primordial magnetic ?elds also depend on the size of the transport coe?cients in the primordial plasma in a crucial way.In particular,the seed?elds created in the QCD phase transition are highly dependent on the neutrino viscosity[6].

Galaxy formation[7]is likewise a?ected by the thermal coe?cients in the primor-dial plasma[8].Viscosity tends to heat up the plasma,and on the other hand thermal conduction transfers heat from regions of high temperature to regions at lower tem-perature.The e?ects a?ect the structure formation and thus in order to make precise models of the galaxy formation it is important to know the values of the transport coe?cients.

It should be noted that two major e?ects are involved when heat transport and viscosity are studied.These e?ects are the number of possible scattering reactions, i.e.the number of particles present in the plasma,and the Debye mass present in the propagator.As we will discuss in Section4,the?rst e?ect will decrease the transport coe?cients while the second e?ect will increase them.The net behaviour of the coe?cients depend on the exact interplay between these two e?ects.

The di?erent interactions give contributions of di?erent strengths to the electrical conductivity as was shown in[17].However,in[17]only strong and electromagnetic interactions were considered.Here we also deal with the scattering reactions involving neutrinos.

The paper is organized as follows.In section2we introduce the reader to the method of calculation and,using the full de?nition of the energy-stress tensor Tμν,we de?ne the transport coe?cients presented here.In section3we reconsider electrical conductivity.We?rst discuss what is the proper way to introduce the Debye screen-ing to the propagators in the matrix elements.We then include additional scattering reactions not present in previous work and present a new value for the electrical con-ductivity.We also compare our results with other recent estimates for the electrical conductivity in the hot plasma.In Section4and5we apply our technique to calcu-lating thermal conductivity and shear viscosity in the hot plasma,respectively,and compare our results with other recent estimates.Section6contains a summary of the results and a brief discussion on their relevance for the early universe.

2The general formalism

2.1The energy-stress tensor

We start by de?ning the transport coe?cients.We do this by using the energy-stress tensor Tμν:

Tμν=(ρ+p)UμUν?pgμν+Tμν

TP +Tμν

EM

,(1)

whereρis the total energy density,p the pressure,Uμthe velocity four-vector(UμUμ=?1)and gμνthe metric tensor.Tμν

EM

is the usual electromagnetic energy-stress tensor

(see e.g.[18])and Tμν

TP

represents the non-ideal contributions from a?nite viscosity and heat conductivity caused by the di?usion of the various particles present in the plasma.

To?nd out the meaning of the transport coe?cients we now follow[9]and derive

the form of Tμν

TP

.Because of dissipation,we must also add a correction term to the particle current Nμ:

Nμ=nUμ+Nμd,(2) where n is the particle number density and Nμd the correction term.To avoid the am-biguities generated by the adding of extra terms induced by dissipation and di?usion, we should de?ne what is meant by the number and energy densities:

n≡?UμNμ,

ρ≡UαUβTαβ,

Uμ≡(?NνNν)?1/2Nμ.(3) Inserting the de?nitions Eq.(3)into Eq.(1)and Eq.(2)we can easily see that the following conditions must be satis?ed:

Nμd=0,

UαUβTαβT P=0.(4) Using the conservation laws for the?uid:

?νTμν=0,

?νNν=0(5) and making use of Eq.(2)and Eq.(4)we?nd that

?νUν=?

Uν

T (?νUμ)Tμν

TP

+

1

Now the task is to construct T μν

TP in the way that the rate of entropy production

per unit volume given by Eq.(7)is positive for all possible ?uid con?gurations.As explained in [9],we may take the dissipative term to be linearly dependent on the space-time derivatives of the four-velocity,energy and number densities.Also,the e?ects considered here are of ?rst order and therefore the adiabatic equations of motion can be used to ?nd out the form of T μν

TP .

It is easiest to continue in the locally moving Lorentz-frame,where U μ=(1,0,0,0).As explained,T μν

TP can now be constructed as a linear combination of the derivatives

?U μ/?x ν,?U μ/?t ,?T/?x νand ?T/?t .Of these,?U 0/?x νvanishes in this frame and ?T/?t can be expressed in terms of ?·U because of adiabacity.Then the most general structure allowed by rotational and space-inversion is

T ij TP =?η ?U i

?x i

?2

?x i

?ξU i

?T )n T ?x γ+U γ

?T

?T )n (g μν+U μU ν)+(?ρ

3g μνU γ+

13δνγU

μ

?T

15 6?(U μU νU γ)?x γ+?U μ?x μ }+O (τ2),(9)

where b is a constant.Since T μν

TP is a tensor,it is su?cient to study it in the locally

comoving Lorentz https://www.doczj.com/doc/f616370022.html,ing once again the properties of this frame,U μ=(1,0,0,0)and ?U 0/?x μ=0,we can somewhat simplify the energy-stress tensor,Eq.(9):

T ij

TP =?4bT 4τ 1?ρ)n 2?·U ?4?x j +?U j

3δij

?·U

+O (τ2),

T i 0TP =4

?x i

+T ?U i

T00

=O(τ2).(10) TP

From Eq.(10)it is easy to extract the values for the transport coe?cients:

ζ=4bT4τ 1?ρ)n 2,

η=(4/15)bT4τand

κ=(4/3)bT3τ.(11) As can be seen from Eq.(11),the transport coe?cients are indeed proportional to the mean free time(or equivalently the mean free path)of particles in the plasma,in which interactions are su?ciently strong so that particles have a high scattering fre-quency.For various species of particles the coe?cients are obviously di?erent because their cross sections vary and thus they have di?erent free paths in the plasma.In Sections4and5we calculate heat conductivity and shear viscosity considering all the particles and their scattering reactions in the hot primordial plasma.We note that the values obtained in Sections4and5are self-consistent with the assumption of the high scattering frequency,i.e.the mean free times estimated from the values of Table 5are very short.Let us?nally note that because the plasma considered here is highly relativistic,and therefore p?1

p0+p i ?f R p0?f?p0E i ??f?p i p0E i=C(p,t)p0,(12)?t

where C(p,t)is the collision integral.Here we have assumed a Robertson-Walker metric with a scale factor R for the background,and we have de?ned F0i=?E i and F ij=?ijk B k.The Robertson-Walker metric is valid here if the mean free path of the particles is su?ciently small i.e.no large?ows are present in the universe. The resulting values in Sections4and5suggest that the usage of this metric is a self-consistent assumption.We also prefer to use co-moving coordinates and de?ne ?f(p,t)= δ(p0?(p2R2+m2)1/2)f(p,p0,t)dp0.(13) Inserting this into Eq.(12),making use of the local momentum de?ned as?p=R p and assuming adiabatic expansion in which E i?(˙R/R)|p|≡H|p|we can use the

Lorentz-frame and write Eq.(12)

in the form (from now on we drop the tildes and we also assume that the electric ?eld is su?ciently small,E i ?T 2)

?f

p 0?f ?p i E i =C (p,t ).(14)

To get a su?ciently good estimate,it is only necessary to consider 2→2reactions,and thus we may use as the collision integral the following expression:

C (p,t )=

1p 0 dP b dP c dP d (2π)4δ4(p +p b ?p c ?p d )

×|M |2([f 0c f 0d [1?f 0b ]+f 0b [1?f 0c ][1?f 0d ])δf ≡?C

(p )δf (p,t ).(17)Eq.(17)assumes that all the particles in the reaction are fermions;generalization to other cases is straightforward.

We have assumed that all the particles b,c and d have an equilibrium distribu-tion.Therefore,if also the test particle has an equilibrium distribution,the collision integral vanishes and only the perturbation term in Eq.(17)survives.Note that the equilibrium distribution depends only on the momentum.

The Boltzmann equation can now be linearized in order to calculate the electrical conductivity.Assuming that δf depends only on p ,the linearized Boltzmann equation reads v i

?f 0?p

i E i =?C (p )δf (p ),(18)where v i is the velocity of the probe particle https://www.doczj.com/doc/f616370022.html,ing Eq.(18)we can now begin to calculate the transport coe?cients in the early universe.

3Eletrical conductivity

There has been many attempts to estimateσ.In[19]this was done in the relaxation time approximation and in[20]and[21]a Coulomb correction was applied.Recently a numerical work was carried out in[17]which showed that the electrical conductivity in the early universe is mainly due to the leptonic contribution and its value for T≤100 MeVσ?0.76T while at T?M W it was found thatσ?6.7T.

However,in[17]the contribution from lepton-quark scatterings was neglected, which was correctly pointed out in[22].We have now included these scatterings and thus acquired a better estimate for the electrical conductivity.The additional reactions now included are of the form l±q→l±q and l±ˉq→l±ˉq,and these a?ect only the leptonic contribution because the quarks interact via the strong interaction and thus the electromagnetic scattering processes mentioned above can still be neglected when considering the electrical conductivity created by the quarks.

The t-and u-channels give singular contributions to the collision integral and that is why one needs regularization.The full?nite temperature calculations with higher order Feynman graphs and resummation would provide the regularization correctly but the expressions for the matrix elements become quite long.Another technical complication is the correct handling of the real and imaginary parts.A suitably accurate calculation can be,however,performed by using thermal masses as regulators in the t-and u-channel propagators.This approach can be expected to give roughly the same results as the one using the full thermal propagator.

The thermal masses used in the t-and u-channels for the lepton,quark,photon and gluon propagators,respectively,are the following[23]:

m2l=

e2

6

T2?0.251T2,

m2γ=

e2

3)T2?1.508(3+

N f

The proper introduction of the thermal regulators in the propagators of the t-and u-channels should also be given special attention.For example,the matrix element for the reaction e?e+→e?e+is the sum(u2+s2)/t2+(u2+t2)/s2+2u2/(st),which in [17]was?rst simpli?ed into the form(s4+t4+u2(s+t)2)/(s2t2)and only after that the regulators were inserted into the t-and u-channels.But one should be extremely careful when applying regularization because the adding of the regulators in di?erent stages of the calculation changes the actual value of the matrix element,and therefore the value of the conductivity.This is so because the regulators are added only to the t-and u-propagators,not consistently everywhere in the matrix element.The proper way to introduce the screening terms is to apply the regulators to every term given by the Feynman graphs separately and before joining the terms together.The regulating should be done this way because the di?erent terms in the matrix elements have singularities of di?erent order and hence the terms should be regulated and integrated separately.

We consider the temperature interval from about1MeV to about10GeV and make the simple approximation that particles appear in the thermal bath when temperature is greater than their mass.The collision integral is performed numerically by evaluating the integral by a simple Monte Carlo simulation.

The results,however,are not very di?erent from the previous ones[17];the quark contribution is still negligible compared with the leptonic electrical conductivity and the actual value of the conductivity is still?10T in natural units.The e?ect of neutrino scatterings to the contibution of either QED or QCD electrical conductivity is negligible.The results are given in Fig.1and in Table5.

The main di?erence between the value for the electrical conductivity here and in[22]is due to the di?erence in the de?nition of the electrical conductivity.In [22]an electric?eld extending itself over the whole universe is used to probe the electrical conductivity of the early universe.Because in[22]a homogenous?eld is assumed to cover the entire space,the leptons(quarks)and antileptons(antiquarks) are accelerated in the opposite directions.In this scenario,it is justi?ed to neglect in the?rst order all other scattering reactions than the lepton-antilepton(quark-antiquark)annihilation.But the resulting electrical conductivity is a global one-one that is the same in the whole universe and does not take into consideration the small patch-like structure of the primordial magnetic?eld.

The primordial magnetic?eld B,if it exists,is most likely random:both its mag-nitude and direction vary.Therefore in the early universe it is more sensible to talk about the local electrical conductivity,σ(x).The magneto hydrodynamic equation reads then

?B

σ(x) .(20)

From Eq.(20)we see that to blindly treat the electrical conductivity strictly as a con-stant would disregard some aspects of the dissipation of the magnetic?eld.However, to a good approximation we can take the electrical conductivity to be a constant inside the patches of the magnetic?eld characterized by correlation length L.

The magnetic?eld is coherent inside the correlation length L.The di?erent patches containing magnetic?eld are uncorrelated and thus a convenient way to de?ne the local conductivity is the measure of the dissipation of the magnetic?elds in the coherent patches.We measure the local electrical conductivity as experienced by a charged particle shot out of one of the magnetic patches to the next,uncorrelated one.The particles are ultrarelativistic because of the high temperature and therefore the di?er-ent bulk velocities between the magnetic patches can be neglected.

The condition when this scenario is valid can be easily checked.The mean free path of a particle in the plasma is l free=(σtp n p)?1,whereσtp is the transport cross section of the particle species p,and n p is its number density.The condition for validity is now that the mean free path is longer than the coherence length of the magnetic?eld, L

However,the coherence length of the primordial magnetic?eld is not known exactly. It might very well be of the order of the natural microphysical scale,the interparticle distance,but there could be processes and phenomena that would cause it to be much larger.For example,one such process studied recently is the inverse cascade[24], which transfers magnetic energy from small length scales to larger length scales.

Here we shall assume that the mean free path of a particle is longer than the correlation length of the primordial magnetic?eld.To calculate the local conductivity, we assume one test particle from each species of particles being shot out of one patch of magnetic?eld to another,totally uncorrelated one.In the new patch,we assume a patch-wide electric?eld to give us a tool to measure the conductivity.Of course,this is just a test?eld,we do not claim such?elds to exist in the early universe.Inside the patch,we also assume isotropy and therefore we can write Eq.(18)to the form

?f0

?e

?f0(p)

?C(p)

d3p,(23)

p

0.0 2.0 4.0 6.0

8.0

0.02.0

4.0

6.0

8.010.0

Electrical conductivity

σ/T

T/GeV QED

QCD

Total

Figure 1:σ/T as a function of temperature.

and conductivity σA ,associated with a given particle species A ,is de?ned by

j A =σA E .

(24)

Thus the contribution of a single species A to conductivity is σA ～1/ |M (AX →Y )|2,where the sum is over all the processes which scatter the test particle A .For the purpose of conductivity,we may view the mixture of di?erent particle species of the early universe a multicomponent ?uid.The ?ow of each component contributes to the total current and adds up to the total conductivity,which reads

σtot = A σA ,(25)

where the sum is over all the relativistic charged species present in the thermal bath.Note that the total conductivity is dominated by the species that has the weakest interaction.This is because the weaker the interaction,the longer time the current ?ow is maintained.

We consider the temperature interval 1MeV <～T <～10GeV,and make the simple,crude assumption that particles appear in the thermal bath only when temperature is greater than their mass.Thus below 100MeV,for example,the only charged particles present are the electrons and positrons.When T >～T QCD ,also the quarks should be counted in.Their main interactions are strong,so that their electromagnetic interactions may be neglected.The list of the relevant reactions involving QED and QCD charged particles can be found in Tables 1and 2.

4Thermal conductivity

To?nd out the thermal conductivity,we now abandon the isotropy requirement and the requirement of locality used in the previous Section in computing the electrical conductivity.Let us for a while let the equilibrium distribution f0depend also on the chemical potentialμ:f0=1/(1±e(p?μ)/T).The conditions for thermal equilibrium require the constancy of temperature and of the sumμ+V,where V is the energy of the particles in an external?eld,throughout the media considered.Here we take V=eφ, whereφis the electric?eld potential.In a plasma with a non-uniform temperature distribution,the electric?eld E is not zero even if the current is zero.In general,when both the current density j and the temperature gradient?T are not zero,the relation between these quantities and the electric?eld can be written as

j=σE?ασ?T+

σ?μ

?p +

p

?r

=?C(p,T)δf.(27)

Assuming that the temperature T and the chemical potentialμare functions of the spatial coordinates it is easy to solveδf,the small perturbation to the equilibrium distribution f from Eq.(27):

δf=p

?p

+

?f0

?T

?T}

=±e(p?μ)/T

pT?C(p,T)

·{e E+?μ+(p?μ)

?T

p

d3p.(29)

Remembering the earlier de?nition ofσ,Eq.(23)and Eq.(24),it is now easy to solve the thermoelectric coe?cient:

α=

1

(1±e(p?μ)/T)2?C(p)

× p2e(p?μ)/T dp

Now let us assume that j=0,in which case we obtain from Eq.(26)thatα?T= E+?μ/e and that

δf=±e(p?μ)/T

T2C(p)

p·?T

3T2 ∞0p3(1±e p/T)2(G?p)dp.(34) The plus-minus sign refers to fermions and bosons,respectively.The heat conductivity coe?cient is,however,always positive because the F(Eq.(16))factor also changes its sign in the collision integral when considering fermions or bosons,respectively.

As explained in Section2,thermal conductivity is related to the mean free time,or equivalently,to the mean free path.The values for the heat conductivity obtained in this Section are self-consistent with the assumption of short mean free times in Section 2as can be seen by comparing the values presented in Table5and Eq.(11).

From Eq.(34)it can be easily seen thatκis proportional to the inverse of the square of the coupling constant of the scattering reaction in question.Thus the smaller the coupling constant is the bigger the value forκwill be i.e.κindeed is related to the mean free path.

The subtle interplay between the increasing e?ect on thermal conductivity by the temperature dependent regulators and the decreasing e?ect due to the change in the amount of particles(potential scatterers)present should also be noted.As can be seen in Fig.2and Table5,the case for the leptonic thermal conductivity,κl,is clear. The?rst e?ect dominates in the lepton case,increasingκl/T2all the way from the case where only electrons are present to the case where also the b-quark is present in the heat bath.The photon case is also straightforward.Because there are no temperature dependent regulators the only e?ect comes from the fact that more and more scatterers are present the higher the temperature is.Thus we see in Fig.2and Table5that the photonic thermal conductivity divided by the temperature squared,κγ/T2,is decreasing steadily as the temperature rises.

The case is similar with gluons and quarks as can be seen in Fig.3and Table 5.The di?erent behaviour of the quark thermal conductivityκq and the gluonic thermal conductivityκg can be understood when one looks at the di?erent processes that are involved in the scatterings.All QCD matrix elements are listed in Table2. The quarks get the biggest contribution to their collision integral from the potentially

0.0 2.0 4.0 6.0

8.0

0.010000.0

20000.0

30000.0

40000.0

Thermal conductivity

κ/T 2

Photons

Charged leptons

T/GeV Figure 2:κ/T 2as a function of temperature for leptons and photons.

singular matrix element which has been regulated by thermal masses while the gluons are not that sensitive to thermal masses.

The thermoelectric coe?cient αin Eq.(30)for leptons is found to be ±39.1(de-pending on the sign of the lepton electric charge)and for quarks 47.4/e q ,where e q is the electric charge of the quark.

The neutrinos do not have any singularities in their matrix elements and thus as the temperature grows and more and more scatterers appear in the heat bath the neutrino thermal conductivity divided by the temperature squared,κν/T 2,decreases steadily.κν/T 2is shown in Fig.4.

Earlier attempts to estimate the heat conductivity can be found,for example,in

[10,12,13].However,in [10]no actual matrix elements were calculated but heat conductivity was estimated to be proportional to the mean free time of a particle species studied.In [12]and [13]a more thorough calculation was performed to the leading logarithmic order.However,due to the approximations used in the formalisms of [12]and [13]we believe that values in the present paper are more accurate.5Viscosity

To ?nd out the shear viscosity in the early universe we,once again,set the stage by starting from the Boltzmann equation Eq.(12)and the de?nition of the shear viscosity and assuming stationary viscous ?ow:

?η(?i V k +?k V i ?2

0.0 2.0 4.0 6.0

8.0

0.05.010.0

15.0

20.0Thermal conductivity

κ/T 2

Gluons

Quarks

T/GeV Figure 3:κ/T 2as a function of temperature for quarks and gluons.0.0 2.0 4.0 6.0

8.0

0.010.0

20.0

30.040.0

Thermal conductivity

κ/(108T 2)

T/GeV νμ and ντ

ντ

νe and

νμ

νe , νμ and ντFigure 4:κ/T 2as a function of temperature for neutrinos.

Table1:Matrix elements|M|2/(32π2αem)for the QED processes used in the collision integral.All matrix elements are summed over?nal spins and averaged over initial spins.αem is the electromagnetic coupling constant,i=j,and e q is the charge of the quark in question.

l?l+→γγ

u2

+1

l?γ→l?γ

u2

+1

l?l+→l?l+

t2+u2+t2

st

u2+s2

u2

+2s2

l i?l j?→l i?l j?

t2

u2+t2

l?l+→qˉq

s2

ut(1

t2

)

3e2q ut(1

t2

)

?us(1

s2

)

?3e2q us(1

s2

)

Table2:Matrix elements|M|2/(16π2αs)for the QCD processes[25]used in the colli-sion integral.All matrix elements are summed over?nal spins and colors and averaged over initial spins and colors.αs is the strong coupling constant,and i=j.

qˉq→gg

3(4

t2

+ut

s2

)?(u2

st

))

2(t2?us

9

(us

s2

)+(u2

st

) 4

t2

+u2+t2

3st

)

4

t2

+s2+t2

3ut

)

4

t2

4

s2

3

9(ut

u2

)?2(s2?ut

us

+t2

gq→gq

t2)?4

u2

+us

ut

+s2

gg→gg

8(17t2?8us

2u2

+17s2?8ut

2ut

+15ts?u2

2su

?135

Table3:Matrix elements|M|2/G2for the neutrino processes used in the collision integral.All matrix elements are summed over?nal spins and averaged over initial spins.G is the Fermi coupling constant,and i=j.For other abbreviations,see Table 4.

νiνi→νiνi

νiνj→νiνj

νiˉνi→νiˉνi

νiˉνj→νiˉνj

νiˉνi→l iˉl i

νiˉνi→l jˉl j

νi l i→νi l i

νi l j→νi l j

νiˉl j→νiˉl j

νiˉl i→νiˉl i

νiˉl i→νjˉl j

νi l j→νj l i

νiˉνi→qˉq

νi q→νi q

νiˉq→νiˉq

νi q1→l i q2

νiˉl i→q1q2

?1

g A

2

1

sin2ΘW

3

1

+2

g q V(charge e q=?1/3)

2

g q A(charge e q=?1/3)

2

0.975

0.221

0.000

0.200

0.979

0.050

whereηis the shear viscosity,v the velocity of a particle and V the velocity of the ?ow.As in Section4,we do not need the requirement of locality used in Section3in the computation of the electrical conductivity.δf can now be deduced from Eq.(18) with E=0:

?f0·v

δf=

?? d3p p i v k v·?(p·V)?f0?C(p)(37)

?C(p)

Let us now assume a?ow V=ay e x,where a is a https://www.doczj.com/doc/f616370022.html,ing V,Eq.(35)and Eq.(37)we can write the following expression for the shear viscosity:

η= d3p v x v y p x p y?f0

2π0dΦ ∞0p4sin2Φcos2Φsin5Θ±e p/T

T π0dΘ

0.0 2.0 4.0 6.08.0

0.010000.0

20000.0

30000.0

Shear viscosity

η/T 3

T/GeV

Photons

Charged leptons

Figure 5:η/T 3as a function of temperature for charged leptons and photons obtained in this Section are self-consistent with the assumption of short mean free times in Section 2as can be seen by comparing the values presented in Table 5and Eq.(11).

Earlier estimates for the shear viscosity in hot (primordial)plasma can be found,for example,in [9,11,12,13,14].However,in [9]no matrix elements were calculated,and it was only noted that the shear viscosity is proportional to the mean free path of a particle species studied.[11,12,13,14]contained much more thorough calculations of the shear viscosity to the leading logarithmic order.The results,however,di?er somewhat from the ones presented in Table 5and Figs.5,6and 7.We believe that the proper introduction and numerical integration of the matrix elements in the collision integral in the present calculation yields more accurate values for ηthan in the earlier publications.

As noted before,in the early universe p ?1

0.0 2.0 4.0 6.0

8.0

0.05.0

10.0

15.0

20.0Shear viscosity

η/T 3Quarks

Gluons

T/GeV Figure 6:η/T 3as a function of temperature for quarks and gluons 0.0 2.0 4.0 6.0

8.0

0.010.020.030.0

Shear viscosity

η/(108T 3

)T/GeV νμ and ντ

ντ

νe and

νμ

νe , νμ and ντFigure 7:η/T 3as a function of temperature for neutrinos.

The way常见用法

The way 的用法 Ⅰ常见用法： 1)the way+ that 2)the way + in which（最为正式的用法） 3)the way + 省略（最为自然的用法） 举例：I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法： 在当代美国英语中，the way用作为副词的对格，“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2）The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3）The way =how/ how much No one can imagine the way he missed her. 4）The way =because

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步骤： （1）计算预期收益率 （2）每个可能的收益率减去预期收益率得到离差 （3）求各离差平方，并将结果与该结果对应的发生概率相乘，然后将这些乘积相加，得到概率分布的方差。 （4）最后，秋初访查的平方根，即得到标准差 4.计算变异系数 变异系数度量了单位收益的风险，为项目的选择提供了更有意义的比较基础。由于变异系数同时反映了风险与收益，股在处理两个或多个具有显著不同预期收益的投资项目时，他是一个更好的风险度量指数。 二、证券组合的风险与收益 1.证券组合收益 证券组合的预期收益，使之组合中单向证券预期收益的加权平均值，权重为整个组合中投入各项证券的资金占总投资额的比重。 2.证券组合的风险 不同于收益，组合风险通常并非组合内部单项资产标准差的加权平均数，事实上，完全可能利用某些风险的单项资产组成一个完全无风险的投资组合。在这一过程中，需要用到统计中的计算相关系数和协方差的知识。 当股票收益完全负相关时，所有的风险都能被分散掉。而当股票收益完全正相关时，则风险无法分散。

The way的用法及其含义(二)

The way的用法及其含义（二） 二、the way在句中的语法作用 the way在句中可以作主语、宾语或表语： 1.作主语 The way you are doing it is completely crazy.你这个干法简直发疯。 The way she puts on that accent really irritates me. 她故意操那种口音的样子实在令我恼火。The way she behaved towards him was utterly ruthless. 她对待他真是无情至极。 Words are important, but the way a person stands, folds his or her arms or moves his or her hands can also give us information about his or her feelings. 言语固然重要,但人的站姿,抱臂的方式和手势也回告诉我们他(她)的情感。 2.作宾语 I hate the way she stared at me.我讨厌她盯我看的样子。 We like the way that her hair hangs down.我们喜欢她的头发笔直地垂下来。 You could tell she was foreign by the way she was dressed. 从她的穿著就可以看出她是外国人。 She could not hide her amusement at the way he was dancing. 她见他跳舞的姿势，忍俊不禁。 3.作表语 This is the way the accident happened.这就是事故如何发生的。 Believe it or not, that's the way it is. 信不信由你, 反正事情就是这样。 That's the way I look at it, too. 我也是这么想。 That was the way minority nationalities were treated in old China. 那就是少数民族在旧中

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(完整版)the的用法

定冠词the的用法： 定冠词the与指示代词this ,that同源,有“那(这)个”的意思,但较弱,可以和一个名词连用,来表示某个或某些特定的人或东西. (1)特指双方都明白的人或物 Take the medicine.把药吃了. (2)上文提到过的人或事 He bought a house.他买了幢房子. I've been to the house.我去过那幢房子. (3)指世界上独一无二的事物 the sun ,the sky ,the moon, the earth (4)单数名词连用表示一类事物 the dollar 美元 the fox 狐狸 或与形容词或分词连用,表示一类人 the rich 富人 the living 生者 (5)用在序数词和形容词最高级,及形容词等前面 Where do you live?你住在哪? I live on the second floor.我住在二楼. That's the very thing I've been looking for.那正是我要找的东西. (6)与复数名词连用,指整个群体 They are the teachers of this school.(指全体教师) They are teachers of this school.(指部分教师) (7)表示所有,相当于物主代词,用在表示身体部位的名词前 She caught me by the arm.她抓住了我的手臂. (8)用在某些有普通名词构成的国家名称,机关团体,阶级等专有名词前 the People's Republic of China 中华人民共和国 the United States 美国 (9)用在表示乐器的名词前 She plays the piano.她会弹钢琴. (10)用在姓氏的复数名词之前,表示一家人 the Greens 格林一家人(或格林夫妇) (11)用在惯用语中 in the day, in the morning... the day before yesterday, the next morning... in the sky... in the dark... in the end... on the whole, by the way...

浅析经济统计学与计量经济学等相关学科的关系及发展前景

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“the way+从句”结构的意义及用法

“tｈｅwａｙ+从句”结构的意义及用法 首先让我们来看下面这个句子: Ｒead the foｌloｗｉnｇpassagｅａnd talkａbout ｉt wi ｔh your classmaｔes．Try tｏｔeｌl whaｔyｏu thiｎk ｏf Tom aｎd ｏｆｔhe ｗay the chiｌｄrenｔrｅated ｈim. 在这个句子中，ｔhe waｙ是先行词,后面是省略了关系副词that或in whｉch的定语从句。 下面我们将叙述“the ｗaｙ+从句”结构的用法。 1.tｈe wａy之后，引导定语从句的关系词是tｈat而不是hoｗ,因此，<<现代英语惯用法词典＞＞中所给出的下面两个句子是错误的：Ｔhｉs is ｔｈeｗayｈowｉtｈａppened. This is the way how he ａlwａｙs treats me. 2．在正式语体中,thaｔ可被in which所代替;在非正式语体中，ｔhat则往往省略。由此我们得到theｗay后接定语从句时的三种模式：1) tｈe ｗaｙ＋ｔhａt-从句２）the way +ｉn whiｃh－从句3) the ｗay +从句 例如：Ｔｈe way(in whiｃh ，thａt) ｔhｅｓｅｃomｒade ｓloｏｋａｔｐrobｌems is wrong．这些同志看问题的方法

不对。 Ｔｈｅｗａy(that ,ｉn ｗhiｃh)yoｕ’ｒe doinｇit is comple ｔeｌy crａzy.你这么个干法,简直发疯。 Wｅａdmiｒｅd him for thｅｗay ｉｎｗhｉch he fａｃeｓdiｆficultieｓ． Waｌlａｃe ａｎd Ｄarwiｎｇreed ｏn the way ｉｎwhi ｃh ｄｉfｆｅrｅnt forms ｏf life had ｂegｕn.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 Thｉs is ｔｈe ｗaｙ（tｈａｔ) hｅdｉd it. I lｉkeｄｔhe waｙ(ｔhat) ｓｈeｏｒganized ｔhe ｍeetｉng． 3.theｗay(tｈat)有时可以与hｏw(作“如何”解）通用。例如： That’s tｈe ｗaｙ(tｈaｔ) shｅspoke. = Tｈat’s how shｅspoke.

数学统计方法在经济学中的应用

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应用，时至今日，数学与经济学相结合，衍生出了数理经济学、经济计量学以及产权经济学等数门专业化理论，经济学中的数学统计方法已经无处不在。将数学方法运用于经济问题的解决中，一般要经历“经济—数学——经济”的模式，既从需要解决的现实经济问题入手，建立数学模型进行，运用数学方法对数学模型进行分析，求得数学结果，再结合经济理论与经济学原理对结果进行评估，得出结论，用于指导经济活动的进行。 2.现代企业经济决策的制定离不开数学统计方法 数学在经济学中的大量运用，使人们对经济活动评估的要求由定性分析发展到定量分析，特别在现代企业在制定决策时，它们都希望通过数学方法来精确的分析决策对企业发展产生的意义。数学方法在现代企业经济决策中的运用，是为了提高经济决策的可靠性与科学性，避免企业财力、物力的损失，通过数学方法对决策执行后的结果进行预测，使企业的发展处于自身可以控制的情况下。一个简单的数学方法就可以将经济决策中的各项因子之间的关系简单的明了的表现出来，各个经济变量之间的关系也能一目了然，经济决策的制定是否可靠的结论就可以得出。作文/zuowen/ 3.数学统计方法是经济理论分析最重要工具之一 数学统计方法是经济学理论分析的最重要工具之一，从最早的代数运用，再到数理经济学中，各种深奥的数学问题中的大量的运用的运用，现代统计经济学中，繁杂数据的中指标的得出，再代现代数学与现代经济理论相结合，产生的特有的专门运用数学方法来解释经济

way 用法

表示“方式”、“方法”，注意以下用法： 1.表示用某种方法或按某种方式，通常用介词in(此介词有时可省略)。如： Do it (in) your own way. 按你自己的方法做吧。 Please do not talk (in) that way. 请不要那样说。 2.表示做某事的方式或方法，其后可接不定式或of doing sth。 如： It’s the best way of studying [to study] English. 这是学习英语的最好方法。 There are different ways to do [of doing] it. 做这事有不同的办法。 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句，但是其后的从句不能由how 来引导。如： 我不喜欢他说话的态度。 正：I don’t like the way he spoke. 正：I don’t like the way that he spoke. 正：I don’t like the way in which he spoke. 误：I don’t like the way how he spoke. 4.注意以下各句the way 的用法： That’s the way (=how) he spoke. 那就是他说话的方式。 Nobody else loves you the way(=as) I do. 没有人像我这样爱你。 The way (=According as) you are studying now, you won’tmake much progress. 根据你现在学习情况来看，你不会有多大的进步。 2007年陕西省高考英语中有这样一道单项填空题： ——I think he is taking an active part insocial work. ——I agree with you_____. A、in a way B、on the way C、by the way D、in the way 此题答案选A。要想弄清为什么选A，而不选其他几项，则要弄清选项中含way的四个短语的不同意义和用法，下面我们就对此作一归纳和小结。 一、in a way的用法 表示：在一定程度上，从某方面说。如： In a way he was right.在某种程度上他是对的。注：in a way也可说成in one way。 二、on the way的用法 1、表示：即将来(去)，就要来(去)。如： Spring is on the way.春天快到了。 I'd better be on my way soon.我最好还是快点儿走。 Radio forecasts said a sixth-grade wind was on the way.无线电预报说将有六级大风。 2、表示：在路上，在行进中。如： He stopped for breakfast on the way.他中途停下吃早点。 We had some good laughs on the way.我们在路上好好笑了一阵子。 3、表示：(婴儿)尚未出生。如： She has two children with another one on the way.她有两个孩子，现在还怀着一个。 She's got five children，and another one is on the way.她已经有5个孩子了，另一个又快生了。 三、by the way的用法

经济学中统计学

《统计学原理》实验指导书 一、实验目的： 1、通过实验让学生了解如何在详细占有有关材料的基础上，灵活应用各种科学的统计分析方法，揭示事物的总体特征、总体变化及其原因，以及揭示事物之间的内在联系及对比关系；从数量的内在联系与对比关系中发现问题、提出问题，然后进行分析研究，从而认识事物的本质和发展规律。 2、掌握用EXCEL进行数据的搜集整理和显示。 二、指导说明 （一）根据调查目的确定调查对象 （二）调查目的及对象确定情况下，该如何设计调查问卷。 （三）操作步骤 1、每位同学根据调查目的先自己设计一份调查问卷。 2、学习委员将该班学生进行分组，选出组长。 3、每一组的组长和小组成员协商，选出一份优秀的调查问卷或将该组所有成员的问卷进行补充，设计出该组的调查问卷。 4、从各组中选出优秀的调查问卷进行比较补充，设计出全班统一的调查问卷。 5、根据调查问卷设计出统一的调查表。 6、各组分工进行资料的收集。 7、资料加工整理。 （1）各组的每个同学先对自己搜集的资料进行加工整理； （2）小组组长汇总该组资料； （3）学习委员汇总各组资料。 8、按照要求运用EXCEL分析资料。即，根据调查表的各项指标或主要指标分析资料。（见以下EXCEL的操作文件） 9、根据分析资料撰写实验报告（或实验总结），字数在1500字左右。 10、将所有书面资料按照实验过程装订。 实验一用Excel搜集与整理数据 一、用Excel搜集数据 二、用Excel进行统计分组 三、用Excel作统计图 实验二用EXCEL计算描述统计量

一、用函数计算描述统计量 二、描述统计工具量的使用 实验三用EXCEL进行时间序列分析 一、测定增长量和平均增长量 二、测定发展速度和平均发展速度 三、计算长期趋势 四、计算季节变动 实验四用EXCEL进行指数分析 一、用Excel计算总指数 二、用Excel计算平均指数 三、用Excel进行因素分析 实验五用EXCEL进行相关与回归分析 一、用EXCEL进行相关分析 二、用EXCEL进行回归分析 实验六用统计软件进行预测 实验七用EXCEL进行参数估计和假设检验 一、用EXCEL进行区间估计 二、用EXCEL进行假设检验 三、实验要求和注意事项 1、预习实验指导书的内容，结合课程所讲的理论，理解实验的目的、内容和操作步骤。 2、上机前准备所需的资料及数据。 3、认真观察实验结果、记录实验结果。 4、对实验结果做简要分析总结。 四、实验成绩评价标准 1、优秀：独立完成实验报告，实验步骤、实验结论正确。 2、良好：实验步骤、结论基本正确。 3、中等：实验步骤、结论出现错误较多。 4、及格：不能独立完成实验报告或步骤、结论出现重大错误，态度不认真。 5、不及格：态度很不认真。

The way的用法及其含义(一)

The way的用法及其含义（一） 有这样一个句子：In 1770 the room was completed the way she wanted. 1770年，这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么？其意义如何？在阅读时，学生经常会碰到一些含有the way 的句子，如：No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中，the way之后的部分都是定语从句。第一句的意思是，“没人知道他是怎样发明这台机器的。”the way的意思相当于how；第二句的意思是，“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中，the way也是as的含义。随着现代英语的发展，the way的用法已越来越普遍了。下面，我们从the way的语法作用和意义等方面做一考查和分析： 一、the way作先行词，后接定语从句 以下3种表达都是正确的。例如：“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如：“火灾如何发生的，有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.

统计学与经济学

统计学与经济学 相关合集：统计学论文 相关热搜：统计学统计学教学统计学应用 对经济统计学的有效性研究是企业提高自身经济效益的重要手段。企业通过统计工作会总结统计信息，统计信息反映的是信息在整个社会经济信息系统中所处的主体地位。 随着统计科学的发展及经济学逐渐走向成熟，经济学与统计学之间相互融合成为了必然。离开了统计分析的任何形式的经济分析，其中包括质和量的分析，实证和理论的分析都是不可能的，所以说统计学对经济发展是有很大的促进作用的，经济学是离不开统计学的支撑的。 1 统计学与经济学的产生及关系 伴随着人类的出现，就出现了结绳记事的统计行为。随着人类的出现就出现了交换的行为，而交换则是当时的经济行为，由此可见人类的统计活动和经济活动是有一样的发展历史的。

随着现代化社会经济的飞速发展，统计学在经济与管理方面发挥着的作用也将逐渐显现出来。统计学将会应用到各个方面，无论是国民经济管理，还是企业管理乃至个人的生产、经营和决策，都要利用统计分析。经济、管理类在现实中更注重实际应用能力。而统计学则是根据实际情况运用各种方法对数据进行分析统计，使得人们通过统计工作对数据有一个直观的认识和了解。我们应该灵活的运用统计学，将统计学与经济学有机的结合起来，为经济发展提供助力。 2 经济统计学的分析与研究 经济统计学在现代经济学中作为一门处理经济数据的科学，经济统计学不但要运用统计方法在经济分析中进行描述和推算，而且要根据实际的经济数据进行统计分析，得出结论并提出改革方案，为企业创造更高的效益。 经济统计学是统计学与经济学相互融合而产生科学结晶，所以我们应该重视经济统计学的发展及利用。在实际生活中如果想要充分利用经济统计学需要很多条件，同时经济统计学需要扎实的数理统计和数学基础作为其发展的支撑。因此我们必须加强数学基础知识的了解，掌握数学统计在经济学中的理论与方法，同时提高数学在经济统计学中的地位，这样才能更好地运用经济统计的方法去解决实际的问题。

way 的用法

way 的用法 【语境展示】 1. Now I’ll show you how to do the experiment in a different way. 下面我来演示如何用一种不同的方法做这个实验。 2. The teacher had a strange way to make his classes lively and interesting. 这位老师有种奇怪的办法让他的课生动有趣。 3. Can you tell me the best way of working out this problem? 你能告诉我算出这道题的最好方法吗？ 4. I don’t know the way (that / in which) he helped her out. 我不知道他用什么方法帮助她摆脱困境的。 5. The way (that / which) he talked about to solve the problem was difficult to understand. 他所谈到的解决这个问题的方法难以理解。 6. I don’t like the way that / which is being widely used for saving water. 我不喜欢这种正在被广泛使用的节水方法。 7. They did not do it the way we do now. 他们以前的做法和我们现在不一样。 【归纳总结】 ●way作“方法，方式”讲时，如表示“以……方式”，前面常加介词in。如例1； ●way作“方法，方式”讲时，其后可接不定式to do sth.，也可接of doing sth. 作定语，表示做某事的方法。如例2，例3；

从统计学的视角看实验经济学

+生产力研究,Ｎｏ．１５．２００８ 从统计学的视角看实验经济学 王维红，顾庆良 （东华大学旭日工商管理学院，上海２０００５１） 【摘要】实验经济学将实验的方法引入经济问题研究，是经济学方法论的一次重大变革，实验方法正逐渐成为经济学研究的一种重要工具。然而，由于实验经济学尚处于发展的初始阶段，对于经济实验的方法还很少有人进行深入的研究。文章指出，实验经济学虽然是一个全新的研究领域，但经济实验本质上是一种统计实验，应遵循统计实验研究的基本方法和基本规律。实验经济学可以借鉴统计学的研究方法，得到更大、更快的发展。 【关键词】实验经济学；信度；效度；实验设计；统计检验 【中图分类号】Ｆ０１１【文献标识码】Ａ【文章编号】１００４－２７６８（２００８）１５－００１０－０３ 实验经济学将实验的方法引入经济问题研究，是经济学方法论的一次重大变革，实验研究方法正逐渐成为经济学研究的一种重要工具。然而，由于实验经济学尚处于发展的初始阶段，对于经济实验的方法还很少有人进行深入的研究，比如实验观测次数的问题、实验结果的信度和效度问题、实验结果的检验问题等。本文将从统计学的视角对这些问题做初步的思考。 一、实验方法在经济学研究方法论上的意义及应用中存在的问题 （一）实验经济学 实验经济学的起源可以追溯到２０世纪３０年代到６０年代，大致可分为三个源头：一是在１９３１年，萨斯通（Ｔｈｕｒｓｔｏｎｅ）首次采用实验的方法来确定个体的无差异曲线；二是在１９５０年决瑟尔和弗鲁德用实验的方法模拟著名的“囚徒困境”问题并得到了与纳什均衡不一致的实验结果。三是哈佛大学的爱德华?张伯伦（ＥｄｗａｒｄＨ．Ｃｈａｍｂｅｒｌｉｎ）等人在１９４２年通过建立一个实验性市场来检验竞争性市场的均衡条件，也得到了与竞争性均衡结果不一致的实验结果。 随后，实验参加者之一史密斯采用双向口头拍卖的集中交易实验方式，发现即使在很少的信息及适度数量的参与者的情况下，市场也能很快地收敛到竞争性均衡，“大量的、具有完美信息的经济代理人”不是市场效率的必要条件。１９６２年史密斯将论文“竞争市场行为的实验研究”发表于权威杂志《政治经济学》，这篇论文被认为是实验经济学诞生的标志，弗农?史密斯也因此被尊称为实验经济学之父。 实验方法是任何学科发展到一定阶段所共有的研究方法，经济学也不例外。实验经济学就是用“受控实验”的方法来研究经济问题。所谓“受控实验”，是指实验具有可控制性和可重复性。虽然用实验方法研究经济问题，可追溯至上百年前甚至更早，但是真正进行经济学“受控实验”距今只有几十年的时间。２００２年，弗农?史密斯获得了诺贝尔经济学奖，这表明实验经济学的研究方法已得到了各国学者的普遍认同。如今，实验研究方法已被广泛应用于行为经济学、产业组织理论、博弈论等许多方面并取得了丰硕的成果。 （二）实验方法在经济学研究方法论上的意义 实验是“科学”实证和发现的最基本也是最有力的工具之一，实验方法对于经济学研究方法论的重要意义在于： １．它表明经济理论的环境和机制基础是可以再造或模拟的，经济理论完全具备实验检验的条件。之前，人们一直认为社会科学是不可实验的。比如诺贝尔经济学奖获得者萨缪尔森在他和诺德豪斯合著的《经济学原理》中说：“经济学家在检验经济法则时，无法进行类似化学家或生物学家的受控实验，因为他们不容易控制其他重 【收稿日期】２００７－０１－２２ 【作者简介】王维红（１９６６－），女，山东济南人，东华大学旭日工商管理学院副教授、博士研究生，研究方向：应用统计学，供应链管理；顾庆良（１９４９－），男，上海人，东华大学旭日工商管理学院教授、博士生导师，研究方向：市场营销。 !"

浅谈统计学在实际生活中的应用

龙源期刊网 https://www.doczj.com/doc/f616370022.html, 浅谈统计学在实际生活中的应用 作者：陈昊 来源：《智富时代》2016年第03期 【摘要】统计学在21世纪的今天生活中变得越来越不可或缺，不论是科技还是美学方面，统计学的概率也在生活中应用的很广泛，基础学科的统计学不仅仅在金融、经济、医学许多领域中运用这统计学的过程，在科技的不断发展在数学中样本的统计也是重要的工具。因此，本文便通过统计学在实际的生活中较大的比重以及大量的重复试验中的随机的统计与概率事件，统计分析是定量与定性的统计工作中的巨大作用的发挥，企业的制定与发展战略生产计划与规划最主要依据。 【关键词】统计学；实际生活；应用；研究 一、统计学的相关论述 统计学对于整理与分析数据以及收集解释相关数据信息的一门科学。统计学是在方法论的性质上的认识科学，统计数据整理和分析的思维产生，发展的提高自身的事物，研究统计的方法与社会科学的性质的统计成果与收集，接近工作的实际，统计的思想主要是变异的思想、均值、相关、拟合的思想的理论研究与逐步系统的形成后的统计理念与统计意识，遵循的指导思想和对研究对象的重要统计学的总体现，“变异”与“一般的水平”，估计的思想则是样本的选取代表了整体的逻辑严谨与必要的预设，相关的思想则是根据哲学的普遍联系的观点指导相关联的总体与个体之间的同质性；拟合思想则是单一的成果是趋势的模型的拟合与预设的基于可能性的关系；最主要的是检验的思想是归纳，也就是验证对于最开始的假设是不是基于局部的特征与规律判断是否完全符合，探索内部的数据数量规律的内在的科学认识，针对的客观之物以及统计事物的步骤即是设计然后进行抽样最后是调查统计并作出推断结论的得出。 二、统计学在各个领域的实际应用 （一）在经济学中的重要应用。统计学的基础知识、数理的统计以及统计分析等在统计学的学习中都是首先要掌握的重要学习项目，这些都是必须要在研究前就必须清晰掌握的基础知识，经济学的分支中的一个统计学的课程的学习。例如，计量的经济学的统计就要依赖这个在金融里面的统计重要意义与地位，金融的计量以及时间的序列是金融和统计的知识的结合，收集、整理的“为何统计”以及“如何进行统计”的思想一直是基础的工具，经济学中主要有两个方面的主要工具性的作用：一是在思想上，统计学的严谨性追求的理性占据着不可小觑的指导与重要的地位的占据经济数据的的描述过程的数据预先处理的方法论是不是科学与实证研究所必须开展的整理与收集；另一个是经济学的研究的最优化的选择的经济研究的约束的条件的经济活动的多样以及研究的错综复杂，研究的成本以及现象的经济研究变得简洁明了。因此，在总体上经济学的统计结论不具有全面的复杂的思想的成本与收益的概念与计算的经济模型与去确定性，并体现了统计学中的经济必然性思想。

the-way-的用法讲解学习

t h e-w a y-的用法

The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.

试从经济学和你的专业角度谈谈你学习统计学的认识

试从经济学和你的专业角度谈谈你学习统计学的认识 在学习统计学之前，一直认为统计学是一门文科方面的课程，只是一些背记的文字和语言之类的。学习之后觉得这是一门比较系统的记录和分析数据的艺术。查了几本关于统计的书本，总结如下： 统计学是应用数学的一个分支，主要通过利用概率论建立数学模型，收集所观察系统的数据，进行量化的分析、总结，并进而进行推断和预测，为相关决策提供依据和参考。统计学主要分为描述统计学和推断统计学。给定一组数据，统计学可以摘要并且描述这份数据，这个用法称作为描述统计学。另外，观察者以数据的形态建立出一个用以解释其随机性和不确定性的数学模型，以之来推论研究中的步骤及母体，这种用法被称做推论统计学。它被广泛的应用在各门学科之上，从物理和社会科学到人文科学，甚至被用来工商业及政府的情报决策之上。 统计学是一门研究随机现象，以推断为特征的方法论科学，“由部分推及全体”的思想贯穿于统计学的始终。具体地说，它是研究如何搜集、整理、分析反映事物总体信息的数字资料，并以此为依据，对总体特征进行推断的原理和方法。用统计来认识事物的步骤是：研究设计—>抽样调查—>统计推断—>结论。这里，研究设计就是制定调查研究和实验研究的计划，抽样调查是搜集资料的过程，统计推断是分析资料的过程。显然统计的主要功能是推断，而推断的方法是一种不完全归纳法，因为是用部分资料来推断总体。统计学是通过数据来进行分析和推断的。因此，统计研究的基础是数据。这些数据的特点是，对于每一个数据而言，都具有不确定性，我们需要抽取一定数量的数据，才可能从中获取信息。因此，统计学的研究依赖于对数的感悟，甚至是对一堆看似杂乱无章的数的感悟。通过对数据的归纳整理、分析判断，可以发现其中隐藏的规律。因为可以用各种方法对数据进行归纳整理、分析判断，所以，得到的结论也可能是不同的。 统计学的分支学科有：理论统计学、统计调查分析理论、经济统计学、社会统计学、卫生统计学、人口统计学、管理统计学、生物统计学、档案统计学等。 非参数检验(括单组资料的非参数检验和两组资料的秩和检验) t检验( t-test) （包括配对t检验和成组t检验）、曼-惠特尼U 检定（Mann-Whitney U）回归分析（regression analysis）（包括一元回归和多元回归）相关性分析（correlation analysis）皮尔森积矩相关系数(Pearson product-moment correlation coefficient) 史匹曼等级相关系数（Spearman's rank correlation coefficient ）卡方分配（chi-square ） 所谓统计思想，就是统计实际工作、统计学理论及应用研究中必须遵循的基本理念和指导思想。统计思想主要包括:均值思想、变异思想、估计思想、相关思想、拟合思想、检验思想。 均值是对所要研究对象的简明而重要的代表。均值概念几乎涉及所有统计学理论，是统计学的基本思想。均值思想也要求从总体上看问题，但要求观察其一般发展趋势，避免个别偶然现象的干扰，故也体现了总体观。 统计研究同类现象的总体特征，它的前提则是总体各单位的特征存在着差异。统计方法就是要认识事物数量方面的差异。统计学反映变异情况较基本的概念是方差，是表示“变异”的“一般水平”的概念。平均与变异都是对同类事物特征的抽象和宏观度量。 估计以样本推测总体，是对同类事物的由此及彼式的认识方法。使用估计方法有一个预设：样本与总体具有相同的性质。样本才能代表总体。但样本的代表性受偶然因素影响，在估计理论对置信程度的测量就是保持逻辑严谨的必要步骤。 事物是普遍联系的，在变化中，经常出现一些事物相随共变或相随共现的情况，总体又是由许多个别事务所组成，这些个别事物是相互关联的，而我们所研究的事物总体又是在同

way的用法总结大全

way的用法总结大全 way的用法你知道多少，今天给大家带来way的用法，希望能够帮助到大家，下面就和大家分享，来欣赏一下吧。 way的用法总结大全 way的意思 n. 道路,方法,方向,某方面 adv. 远远地，大大地 way用法 way可以用作名词 way的基本意思是“路,道,街,径”,一般用来指具体的“路,道路”,也可指通向某地的“方向”“路线”或做某事所采用的手段,即“方式,方法”。way还可指“习俗,作风”“距离”“附近,周围”“某方面”等。 way作“方法,方式,手段”解时,前面常加介词in。如果way前有this, that等限定词,介词可省略,但如果放在句首,介词则不可省略。

way作“方式,方法”解时,其后可接of v -ing或to- v 作定语,也可接定语从句,引导从句的关系代词或关系副词常可省略。 way用作名词的用法例句 I am on my way to the grocery store.我正在去杂货店的路上。 We lost the way in the dark.我们在黑夜中迷路了。 He asked me the way to London.他问我去伦敦的路。 way可以用作副词 way用作副词时意思是“远远地,大大地”,通常指在程度或距离上有一定的差距。 way back表示“很久以前”。 way用作副词的用法例句 It seems like Im always way too busy with work.我工作总是太忙了。 His ideas were way ahead of his time.他的思想远远超越了他那个时代。 She finished the race way ahead of the other runners.她第一个跑到终点,远远领先于其他选手。 way用法例句