Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki...

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1 Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 02, 2011) Johnson noise Johnson-Nyquist theorem Boltzmann constant Parseval relation Correlation function Spectral density Wiener-Khinchin (or Khintchine) Flicker noise Shot noise Poisson distribution Brownian motion Fluctuation-dissipation theorem Langevin function ___________________________________________________________________________ John Bertrand "Bert" Johnson (October 2, 1887–November 27, 1970) was a Swedish-born American electrical engineer and physicist. He first explained in detail a fundamental source of random interference with information traveling on wires. In 1928, while at Bell Telephone Laboratories he published the journal paper "Thermal Agitation of Electricity in Conductors". In telecommunication or other systems, thermal noise (or Johnson noise) is the noise generated by thermal agitation of electrons in a conductor. Johnson's papers showed a statistical fluctuation of electric charge occur in all electrical conductors, producing random variation of potential between the conductor ends (such as in vacuum tube amplifiers and thermocouples). Thermal noise power, per hertz, is equal throughout the frequency spectrum. Johnson deduced that thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise. http://en.wikipedia.org/wiki/John_B._Johnson

Transcript of Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki...

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Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton (Date: January 02, 2011)

Johnson noise Johnson-Nyquist theorem Boltzmann constant Parseval relation Correlation function Spectral density Wiener-Khinchin (or Khintchine) Flicker noise Shot noise Poisson distribution Brownian motion Fluctuation-dissipation theorem Langevin function ___________________________________________________________________________ John Bertrand "Bert" Johnson (October 2, 1887–November 27, 1970) was a Swedish-born American electrical engineer and physicist. He first explained in detail a fundamental source of random interference with information traveling on wires. In 1928, while at Bell Telephone Laboratories he published the journal paper "Thermal Agitation of Electricity in Conductors". In telecommunication or other systems, thermal noise (or Johnson noise) is the noise generated by thermal agitation of electrons in a conductor. Johnson's papers showed a statistical fluctuation of electric charge occur in all electrical conductors, producing random variation of potential between the conductor ends (such as in vacuum tube amplifiers and thermocouples). Thermal noise power, per hertz, is equal throughout the frequency spectrum. Johnson deduced that thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise.

http://en.wikipedia.org/wiki/John_B._Johnson

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____________________________________________________________________________ Harry Nyquist (February 7, 1889 – April 4, 1976) was an important contributor to information theory.

http://en.wikipedia.org/wiki/Harry_Nyquist ___________________________________________________________________________ 10S.1 Histrory

In 1926, experimental physicist John Johnson working in the physics division at Bell Labs was researching noise in electronic circuits. He discovered random fluctuations in the voltages across electrical resistors, whose power was proportional to temperature. Harry Nyquist, a theorist in that division, got interested in the phenomenon and developed an elegant explanation based on fundamental physics. hence this type of noise is called Johnson noise, Nyquist noise, or thermal noise. 10S.2 Spectral density

Suppose we measure a physical quantity x(t) as a function of time t. We define the Fourier transform of )(tx  as

)(2

1)( txdteX ti

Note that

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)()(2

1)(*

XtxdteX ti .

since x(t) is real. The inverse Fourier transform is given by

)(2

1)(

Xedtx ti .

(a) Parseval relation.

0

2

2

*

)'(*

*'*

)(2

)(

)'()'(')(22

1

)'(')(2

1

)'('2

1)(

2

1)()(

Xd

Xd

XdXd

dteXdXd

XedXeddtdttxtx

ti

titi

since

2*2

)()()()()()( XXXXXX .

(b) We use a truncated signal.

It is sometimes necessary, in order to avoid convergence problem, to approach the definition of the integral as a limiting process. Using the Parseval relation, we get

0

22/

2/

22 )(2

lim)(1

lim)( XdT

dttxT

txT

T

TT

.

We define the spectral density as

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)()(2

lim2 GX

TT

, (1)

which is the ensemble average of 2

)(X . Then we have

0

2 )()( Gdtx . (2)

((Note)) We assume that the average of process parameter over time and the average over the statistical ensemble are the same (ergodicity). 10S.3 Correlation function

The limiting process required to calculate G() is sometimes awkward to use in practice. Here we show that correlation function can be closely related to the spectral density. The correlation function associated with a stationary random function of time t, x(t), is defined by

txtx ()( . (3)

which means that Eq.(3) is independent of t. The Fourier transform of )( tx is given by

)()'(2

1)(

2

1 '

Xetxdteetxdte itiiti

.

Correspondingly, the inverse Fourier transform is given by

)(2

1)( )(

Xedtx ti .

We need to calculate the product given by

)'()('2

1)()( ')(

XXeeddtxtx iti .

It is sometimes necessary, in order to avoid convergence problem, to approach the definition of Eq.(3) as a limiting process. The integral of this product is

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0

2

2

'

2/

2/

')'(2/

2/

)()cos(2

)(

)()(

)'()()'('22

1

)'()('2

1)()(

Xd

Xed

XXed

XXedd

XXedtedddttxtx

i

i

i

T

T

itiT

T

Note that

)'(22/

2/

)'(

T

T

tidte .

Then we have

0

2

2/

2/

)()cos(2

lim

)()(1

lim()(

XdT

dttxtxT

txtx

T

T

TT

The correlation function C() is obtained as the ensemble average of txtx ()( ,

)(2

1

)()cos(

)()cos(2

lim

()()(

0

0

2

Ged

Gd

XdT

txtxC

i

T

, (4)

where we use the relation(Eq.(1)). The inverse Fourier transform of G() is given by

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2

)(2)(

2

1 CGed i

.

The corresponding Fourier transform is

0

)()cos(2

4

)(2

2

2

)(2

2

1)(

Cd

Ced

CedG

i

i

. (5)

Equation (5) is the Wiener-Khinchin (WK) theorem. G() and C() are a Fourier transform pair;

G(): spectral density

C(): correlation function 10S.4 Properties of the correlation function

The correlation function txtxC ()()( is an even function of .

)()()()()(()()( CtxtxtxtxtxtxC .

We also have

0)]()([ 2 txtx ,

or

0)()(2)()( 22 txtxtxtx ,

or

)()()()()0( 2 CtxtxtxC . 

10S.5 Johnson-Nyquist theorem

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At any non-zero temperature we can think of the moving charges as a sort of electron gas trapped inside the resistor box. The electrons move about in a randomised way — similar to Brownian motion — bouncing and scattering off one another and the atoms. At any particular instant there may be more electrons near one end of the box than the other.

We consider a resistance R which has a length l, a cross section A. The relaxation time of the

carriers (electrons) is e. There are n electrons per unit volume (n is called the electron density). The total number of electrons is N;

nAlN . From the Ohm's law, the open circuit voltage is

)( uneRARJAIRV ,

where V is a voltage, I is a current, J is the current density, A is the conductor cross section area, e is the absolute value of the charge of electron (e>0), and u  is the average velocity of electrons. Since

N

iiu

Nu

1

1,

we have

N

ii

N

ii

N

ii Vu

l

eRu

Ne

Al

NRAV

111

1)( ,

where

ii ul

eRV .

Suppose that the correlation function is expressed by

00 22

2()()(

eu

l

eReVtVtVC iiii .

The spectral density is given by

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20

022

0

22

0

)(12

4

)cos(2

4

)()cos(2

4)(

0

i

i

ul

Re

edul

Re

CdG

(Lorentz-type spectral density)

In general, the spectral density G() has a Lorentz-type when the correlation function C() is

described by a exponential decay; exp(-/t0). For the voltage signals, the spectral density G() is customary to use units of [V2 s] = [V2 Hz−1].

We note that

Tkum Bi 2

1

2

1 2 , (equipartition theorem)

where m is the mass of electron and kB is the Boltzmann constant. In metal at room temperature,

e is very small; e<10-12 s. For the DC and microwave regions, 12 cf . Then we get

0

2Re

2

4)(

m

Tk

lG B

, (white noise)

which is independent of . We get

0

2

0

2

4

)(2

)()(

m

Tk

l

Ref

Gf

GGdV

B

i

since = 2f. We have

fTRkml

NAeRfTRkVNV BBi 4)(4

20

222

.

Here we note that the resistance is define by

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02

2

02

1

NAe

ml

Am

eAlN

l

A

lR ,

and

12

02

02

2

20

2

ml

NAe

NAe

ml

ml

NAeR

.

Experimentally it is predicted that the plot of 2V vs R may exhibits a straight line. The slope

will give some estimation for the value of the Boltzmann constant kB.

10S.6 Evaluation of 2V  

The Boltzmann constant is given by

kB = 1.380650410 x 10-23 J/K.

fTRkV B 42 ,

where T is the temperature (K), R is the resistance, and f is the frequency range.

(i) T = 293.15 K, f = 106 Hz, R = 1 M

VV 238.1272 .

(ii) T = 293.15 K, f = 1.9980 x 104 Hz, R = 200

VnVV 254.0349.2542 .

(iii) T = 293.15 K, f = 1.0 x 104 Hz, R = 1000

VnVV 402.0362.4022 .

10S.7 Equivalent circuit (Thevinin theorem and Norton theorem)

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Noise volatge source (the left of the above figure) and noise current source (the right of the above figure);

RG

1

fTRkV B 42

.

)(4

422

2

2 fTGkR

fTRk

R

VI B

B

.

In Fig. we show the equivalent circuits for the noise voltage source and noise current source. These circuits are equivalent according to the Thevinin's theorem and Norton's theorem. ________________________________________________________________________ 10S.8 Measurement of Johnson noise

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The above figure shows an equivalent circuit for the Johnson noise produced by a resistor. n is an invisible random voltage generator, connected in series with an ideal (noise free) resistor R. The voltage fluctuations are amplified and passed through a band-pass filter to a voltmeter. The filter only allows through frequencies in some range (the bandwidth),

minmax fff , (Hz)

Rin is the input resistance of the amplifier. The voltage due to the Johnson noise is

fTRkVe Bn 42 .

From the Ohm's law, the current flowing in the input resistance of the amplifier is

in

n

RR

ei

.

The voltage seen at the amplifier's input across Rin is

in

n

RR

ReRiv

.

The noise power entering in the amplifier is

22

in

n

RR

ReivP

.

The maximum noise power is obtained as

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)(44

22

max fTkR

V

R

eP B

n ,

when Rin = R (impedance matching). The unit of Pmax is Watt (J/s) 10S.9 Shot noise

Shot noise is distinct from current fluctuations in thermal equilibrium, which happen without any applied voltage and without any average current flowing. These thermal equilibrium current fluctuations are known as Johnson-Nyquist noise or thermal noise. Shot noise is a Poisson process and the charge carriers which make up the current will follow a Poisson distribution. We define the Fourier transform of the current,

)(2

1)( tIdteY ti

.

Then we have

0

22/

2/

22 )(2

lim)(1

lim YdT

dttIT

IT

T

TT

.

We define the spectral density as

)()(2

lim2 SY

TT

, (1)

which is the ensemble average of 2

)(Y . Then we have

0

2 )(SdI .

The correlation function of current is defined by

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)(2

1

)()cos(

)()cos(2

lim

()()(

0

0

2

Sed

Sd

YdT

tItIC

i

T

or

2

)(2)(

2

1 CSed i

.

The Wiener-Khinchin (or Khintchine) states that the noise spectrum is the Fourier transform of the correlation function (the spectral density)

0

)()cos(2

4

)(2

2

2

)(2

2

1)(

Cd

Ced

CedS

i

i

((Note))

S(): spectral density

C(): correlation function The delta-function current pulses are given by

k

kttetI )()( , k

kttetI )()( .

Then the correlation function is obtained as

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)(

)(lim

)()(lim)(

''

2

'

2/

2/

'

2

Ie

ttT

e

dtttttT

eC

k kkk

T

k k

T

T

kkT

where

e

I

T

N ,

and I is the DC current. When the summation indices are k = k', it means that the arrival times

are equal tk = tk'. Then we just have (). If there are N values of tk such that -T/2<tk<T/2, these

terms will contribute N() to the correlation function. For tk ≠ tk' the delta functions will occur

at randomly distributed, nonzero values of . The contributions from these delta functions to the

C() will vanish. Taking the Fourier transform, we find

IeedIeS i 22

1)(

2

2)(

The spectrum is uniform and extends to all frequencies. Such a spectrum is called white. Then we have

fIeIef

IedSdI

222

22

2

1)(

00

2

The current fluctuations have a standard deviation of

fIeI 22 ,

where e (>0) is the absolute value of the charge of electron, Δf is the bandwidth in hertz over which the noise is measured, and I is the average current through the device. For a current of 100 mA this gives a value of

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nAI 179.02

if the noise current is filtered with a filter having a bandwidth of f = 1 Hz. The plot of 2I vs

I may exhibits a straight line. This slope will give some estimation for the value of charge e.

e = 1.602176487 x 10-19 C (from NIST Physics constant) ((Note)) Derivation of the formula by van der Ziel (Poisson distribution)

We define N as the number of carriers passing a point in a time T at a rate n(t).

T

dttnN0

)( , and TnN .

where N and n are ensemble averages and this result follows from that fact that time average equals ensemble average (the ergodicity). We assume that n follows the Poisson statistics. The average current is

neT

NeI

The spectral density S is given by

Iee

IeneneS 2222 2222

where nn 2 for the Poisson distribution. Since S is independent of the frequency f (white noise), we have

fIefSI 22

10S.10` Flicker noise (1/f noise, pink noise): DC current related noise

Flicker noise, also known as 1/f noise, is a signal or process with a frequency spectrum that falls off steadily into the higher frequencies, with a pink spectrum. It occurs in almost all electronic devices, and results from a variety of effects, though always related to a DC current.

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Fig. Spectral density vs frequency. 1/f noise in the low frequency range (the flicker noise).

White noise in the higher frequency range. ______________________________________________________________________________ 10S.11 Brownian motion

Suppose that a force (t) is applied to a particle with a mass m along the x direction. According to the Newton's second law, The velocity )()( txtv satisfies the following

differential equation,

)()()(

ttvdt

tdvm . (1)

In a situation such that the particle moves randomly in the fluid at a constant temperature T, the

parameters and (t) no longer be regarded as independent ones. It is required that and (t) are closely related to each other. This is the key point of the Brownian motion (Einstein's relation).

Note that (t) is a force applied to the particle as a result of the collision of molecules of fluid with the particle. This force is considered to be random force (fluctuating force). We assume that

the time average of (t) is zero and exhibits a white random force (Gaussian);

0)( t , and

)'(2)'()( tttt , (2)

where is the magnitude of the random force. The first term of Eq.(1) is the effect of friction,

where is the coefficient of friction. We define the relaxation time as

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m .

The solution of Eq.(1) is obtained as

t

t

sttt sdsem

tvetv0

0 )(1

)()( /)(0

/)( .

When )( 0tv is fixed,

)()(1

)()( 0/)(/)(

0/)( 0

0

0 tvesdsem

tvetv ttt

t

sttt ,

indicating that )(tv decays with time (relaxation time ). For simplicity, we assume that v(t0) is

finite. In the limit of t0 →-∞, we have

t

st sdsem

tv )(1

)( /)( .

Then we get

0)(1

)( /)(

t

st sdsem

tv ,

since 0)( s . The correlation function is given by

0

21212

0

12

21212

21212

2211221

)(2

)(2

)()(1

)()(1

)()(

21

2 221 11

2 221 11

2 221 11

uuttdueduem

ssdsedsem

ssdsedsem

dssedssem

tvtv

uu

t stt st

t stt st

t stt st

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where t1 - s1 = u1 and t2 - s2 = u2. Using the Mathematica, we get

||

221

21

)()(tt

em

tvtv

. (3)

_____________________________________________________________________________ ((Mathematica))

_____________________________________________________________________ When t = t', we have

22 )(

mtv

. (4)

Using the equipartition law in the classical limit,

Tktvm B2

1)(

2

1 2 , (5)

we get the relation

22 )(

mm

Tktv B

, (6)

or

eq1

Integrate

IntegrateExpu1 u2

DiracDeltat1 t2 u1 u2,

u1, 0, , u2, 0, Simplify, t1 t2 Reals & ;

Simplifyeq1, t1 t21

2t1t2

Simplifyeq1, t1 t21

2

t1t2

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TmkB . (7)

This is the relation of dissipation-fluctuation. The parameter is the magnitude of the fluctuating force and /m is the co-efficient of friction (dissipation). This relation is first derived by

Einstein (1905). 10S.12 Diffusion constant D

The velocity correlation function is rewritten as

||

21

21

)()(tt

B em

Tktvtv

.

The displacement correlation function can be obtained as follows.

)}1({2

)()()]0()([

/2

0

||

2

0

1

0

212

0

12

21

tB

t tttB

tt

etm

Tk

edtdtm

Tk

tvtvdtdtxtx

When t>>, we have

tTk

tm

m

Tkt

m

Tkxtxx BBB

222

)]0()([ 22 . (8)

We define the Diffusion constant D as

Tk

t

xtxD B

t

2

)]0()([lim

2

. (9)

((M athematica))

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10S.13 Langevin function

We consider the case that the finite force F is further applied to the particle. We set up the Lagrange equation

Fttvdt

tdvm )()(

)( . (10)

Using the assumaption that 0)( t , we get

Ftvdt

tvdm )(

)( . (11)

The solution of this equation is

00

00

)0()1(

)(]1[)()(

0

)(

tttt

ttm

ttm

evem

F

etveF

tv

In the limit of t→∞ (in thermal equilibrium), )(tv becomes constant (steady state),

Fm

Ftv

)( , (12)

where is the mobility and is defined by

Clear"Global`";

f1 Expt1 t2

UnitStept1 t2

Expt1 t2

UnitStept2 t1;

eq1

IntegrateIntegratef1, t1, 0, t,

t2, 0, t Simplify, t 0, 0 &

2 t 1 t

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m

. (13)

There is a relation between D and as

Tkm

TkD B

B , (Einstein's relation), (14)

10S.14 Fluctuation-dissipation (FD) theorem (i) First-type FD theorem

The velocity correlation function is given by

||

)0()(t

B em

Tkvtv

.

Taking the integral over time, we have

Tkm

Tkdte

m

Tkdtvtv B

Bt

B

0

|

0

)0()( . (15)

Then the mobility can be rewritten as

0

)0()(1

dtvtvTkB

. (Type-1 FD theorem) (16)

The transport co-efficient (mobility, conductivity) can be described by the time correlation of the velocity (current density). (ii) Second-type FD theorem

From the relation

)(2)0()( tt ,

We have

Tkdttdtt B 22)(2)0()(

,

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or

Tkdttdtt B 22)0()(2)0()(0

,

or

dttTkB

)0()(1 . (17)

The co-efficient of the friction can be described by the time correlation of the fluctuating force. 10S.15 Langevin equation for electrical conductivity

We consider the motion of the i-th particle with mass m and charge e in the presence of

fluctuating electric field i(t);

eEtetvtvdt

dm iii )()](

1)([

We define the current density as

n

ii tevtJ

1

)()(

From the above equation, we have

neEtetvtvdt

dm

n

ii

n

ii

n

ii

111

)(])(1

)([

,

or

EnetnetJtJdt

dm 22 )()](

1)([

,

where n is the particle density and the fluctuating electric field is given by

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n

ii t

nt

1

)(1

)(

which is a average fluctuating electric field applied to the n charged particles.

0)( t .

and

)'(2

)'()(2 ttTmk

ttq Bijji

.

Here we assume the independence of { )(ti }. In the steady state, we have

EEm

netJ

2

)( .

The electrical resistance R is

Anq

ml

A

l

A

lR

2

1

The fluctuating electric field (t) can be described by

)'(2)'(2)'(2

)'()(2

ttl

RATktt

ml

RATmktt

nq

Tmktt BB

B

Using the relatrion V(t) = l (t),

)'(2)'()( ttTRAlktVtV B .

or

)(2)0()( tTRAlkVtV B

or

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24  

0

)0()(11

)0()(2

11dtVtV

TklAdtVtV

TklAR

BB

10S.16 Fluctuation-dissipation theorem for Conductivity

Suppose that E = 0. Then we have

)()(1

)(2

tm

netJtJ

dt

d

The solution for this equation is given by

0

0

)()()( 0

)(2 ttt

t

st

etJdssem

netJ

For simplicity, we assume that J(t0) is finite. In the limit of t0 →-∞, we have

t st

sdsem

netJ )()(

)(2

.

Then we get

0)()()(2

t st

sdsem

netJ ,

since 0)( s . The correlation function is given by

0

21212

0

1

2

212122

42

21212

42

22112

42

21

)(2

)(2

)()(

)()()()(

21

2 221 11

2 221 11

2 221 11

uuttdueduem

Tkne

ssdsedsene

Tmk

m

en

ssdsedsem

en

dssedssem

entJtJ

uuB

t stt stB

t stt st

t stt st

where t1 - s1 = u1 and t2 - s2 = u2. Using the Mathematica, we get

Page 25: Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki ...bingweb.binghamton.edu/~suzuki/Math-Physics/LN10S_Jphnson-Nyquist_noise.pdfacross electrical resistors, whose power was proportional

25  

2121 22

21 2

2)()(

tt

B

tt

B em

Tknee

m

TknetJtJ

.

Then we get

Tkm

Tknedte

m

TknedtJtJ B

Bt

B

2

0

2

0

)0()( .

The conductivity is expressed by the time correlation of the current dnsity;

0

)0()(1

dtJtJTkB

.

_____________________________________________________________________________ REFERENCES A.C. Melissinos and J. Napolitano, Experiments in Modern Physics, 2nd edition (Academic

Press, Amsterdam, 2003). J.B. Johnson, Phys. Rev. 32, 97 (1928). H. Nyquist, Phys. Rev. 32, 110 (1928). C. Kittel, Elementary Statistical Physics (John Wiley & Sons, New York, 1958). C. Kittel and H. Kromer, 2nd edition (W.H. Freeman and Company, New York, 1980). H.S. Robertson, Statistical Thermodynamics (PTR Prentice Hall, Englewood Cliffs, NJ, 1993). Toshimitsu Musha, World of Fluctuation (Kozansha Blue Backs, Tokyo, Japan) [in Japanese]. Masuo Suzuki, Statistical Mechanics 2nd edition (Iwanami, Tokyo, 1996) [In Japanese]. Kazuo Kitahara, Nonequilibrium Statistical Physics, 2nd edition (Iwanami, Tokyo, 1998) [in

japanese]. __________________________________________________________________________ APPENDIX A.1 Poisson distribution The Poisson ditribution function is given by

!n

eP

n

n

,

with the mean

Page 26: Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki ...bingweb.binghamton.edu/~suzuki/Math-Physics/LN10S_Jphnson-Nyquist_noise.pdfacross electrical resistors, whose power was proportional

26  

0nnnPn ,

and the variance

0

2222 )(n

nPnnn ,

since

)1(0

22

n

nPnn .

Fig. The Poisson distribution function with being changed as a parameter. ((Mathematica))

m=10

20

3040

5060 70 80 90 100

20 40 60 80 100 120n

0.02

0.04

0.06

0.08

0.10

0.12

Poisson

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27  

PDFPoissonDistribution, k k

k

MeanPoissonDistribution

VariancePoissonDistribution

f1

ListPlotTablek, PDFPoissonDistribution10, k,

k, 0, 30;

f2

GraphicsTextStyle"10", Black, 12,

10, 0.07;

Showf1, f2

m=10

5 10 15 20 25 30

0.02

0.04

0.06

0.08

0.10

0.12

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28  

A.2 Gaussian distribution (normal distribution)

]2

)(exp[

2

1),;( 2

2

x

xf .

dxxxfx ),,( .

22222 ),,()(

dxxfxxx .

: mean

: standard deviation

])(2

1exp[]

2

)(exp[),;(2 2

2

2

xx

xf .

s=0.1

0.2

0.30.40.50.6

-1.0 -0.5 0.5 1.0x-m

1

2

3

4

Gaussian

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29  

The full width at half maximum (FWHM);

35482.2

17741.12ln2 ((Mathematica))

1

2

2 ln2- 2 ln2

FWHM

-4 -2 2 4

x - m

s

0.2

0.4

0.6

0.8

1.0

f 2p s

Page 30: Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki ...bingweb.binghamton.edu/~suzuki/Math-Physics/LN10S_Jphnson-Nyquist_noise.pdfacross electrical resistors, whose power was proportional

30  

PDFNormalDistribution, , x

x2

2 2

2

MeanNormalDistribution,

VarianceNormalDistribution, 2

f1 PlotPDFNormalDistribution0, 1, x,

x, 4, 4;

f2

GraphicsTextStyle"0, 1", Black, 12,

1, 0.3;

Showf1, f2

m=0, s=1

-4 -2 2 4

0.1

0.2

0.3

0.4