Gaussian integral Consider the multivariable Gaussian
distribution, P(x) det A (2) n 1/2 exp 1 2 xi Aij xj , (1.77)
where A is a positive definite matrix of rank n. A mathematical
result which is extremely important throughout physics is the
following: Z(
p p0 , where p0 is the ambient pressure at the top of the
reservoir: p p0 = g (h2 h1 ) , (2.7) where g is the acceleration
due to gravity. The height h1 of the left column of fluid in the Utube provides a measure of the change in the volume of the
gas: V
equations, we find VB VA = VC VD , (2.83) and therefore W =
R(T2 T1 ) ln VB VA (2.84) QAB = RT2 ln VB VA . (2.85)
Finally, the efficiency is given by the ratio of these two
quantities: = W QAB = 1 T1 T2 . (2.86) 2.6.5 The Stirling cycle
Many other engine
heat, Q1 , with 0 < Q1 < Q2 , is deposited in a second reservoir
at a lower temperature T1 . Its temperature change T1 =
+Q1/C1 is also negligible. The difference W = Q2 Q1 is
extracted as useful work. We define the efficiency, , of the
engine as the rati
someone chosen at random from the quarantine group actually
has the disease? We use Bayes theorem with the binary
partition cfw_A, A. Let B denote the event that an individual tests
positive. Anyone from the quarantine group has tested positive.
Given thi
pA exerted by the gas, is reversible. To reverse this process, we
simply add infinitesimally more force to pA and the gas
compresses. A quasistatic process which is not reversible:
slowly dragging a block across the floor, or the slow leak of air
from a t
angry with me: my father and my dog. Figure 1: My father
(Louis) and my dog (Henry). xiv CONTENTS 0.2 General
references L. Peliti, Statistical Mechanics in a Nutshell
(Princeton University Press, 2011) The best allaround book on
the subject Ive come acr
ln q = N h 1+x 2 ln 1+x 2 + 1x 2 ln 1x 2 i + N h 1+x 2 ln p +
1x 2 ln q i . (1.12) Notice that the terms proportional to N ln N
have all cancelled, leaving us with a quantity which is linear in
N. We may therefore write ln PN,X = Nf(x) + O(ln N), where f(
the state variables are the same at the beginning and the end,
we have Ecyclic = Q W = 0 = Q = W (cyclic) . (2.23) 2.5. THE
FIRST LAW OF THERMODYNAMICS 27 2.5.2 Single component
systems A single component system is specified by three state
variables. In m
occur with probabilities cfw_pn . What is the expected amount of
information in N observations? Since event n occurs an average
of Npn times, and the information content in pn is ln pn, we
have that the average information per observation is S = hIN i N
=
work done by the engine, respectively, and Q and W to denote
the heat taken from the engine and the work done on the
engine. A perfect engine has Q1 = 0 and = 1; a perfect
refrigerator has Q1 = Q2 and = . Both violate the Second
Law. Sadi Carnot7 (1796 18
hx 2 i hxi 2 = 2 . (1.64) We call the mean and the
standard deviation of the distribution, eqn. 1.62. The quantity
P() is called the distribution or probability density. One has
P() d = probability that configuration lies within volume d
centered at For e
FRMULAS PARA LIQUIDAR
ACREENCIAS LABORALES
Por pasos:
1. OBTENCIN DEL TIEMPO DE SERVICIO:
Un trabajador ingres el 7 de abril de 2013 y renunci el 10 de marzo de 2015.
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MESES
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2013
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AOS:
MESES:
DAS:
03
04
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2 x 360
1 x 30
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=
=
=
Figure 1.2: Comparison of exact distribution of eqn. 1.10 (red
squares) with the Gaussian distribution of eqn. 1.19 (blue line).
1.2.3 Entropy and energy The function f(x) can be written as a
sum of two contributions, f(x) = e(x) s(x), where s(x) = 1+x 2
the left at level j. At each level, the probability for these two
outcomes is given by P = p ,+1 + q ,1 = ( p if = +1 q if
= 1 . (1.2) This is a normalized discrete probability distribution
of the type discussed in section 1.4 below. The multivariate
dis
the gas consists of constituent atoms or molecules, which are
constantly undergoing collisions with each other and with the
walls of the container. When a particle bounces off a wall, it
imparts an impulse 2n(n p), where p is the particles
momentum and n
quantity, scaling with the size of the system. If we divide by the
number of moles N/NA, we obtain the molar heat capacity,
sometimes called the molar specific heat: c = C/, where =
N/NA is the number of moles of substance. Specific heat is also
sometimes
n ) = 1 Z exp ( X K a=1 a Xa n ) , (1.43) where Z = e 1+0 is
determined by normalization: P n pn = 1. This is a (K + 1)parameter distribution, with cfw_0 , 1 , . . ., K determined by
the K + 1 constraints in eqn. 1.41. Example As an example,
consider the
and the total number of such sequences will be g(N) = N! QK
n=1 Nn! . (1.26) In general, this is far less that the total possible
number KN , and the number of bits necessary to specify one
from among these g(N) possibilities is log2 g(N) = log2 (N!) X K
fell down a flight of stairs. This aggravated a developing heart
condition, which he refused to treat with anything other than
more heat. Two weeks later, he died. Fouriers is one of the 72
names of scientists, engineers and other luminaries which are
eng
j3 j4 + A 1 j1 j3 A 1 j2 j4 + A 1 j1 j4 A 1 j2 j3 . (1.81) 1.5
Appendix : Bayesian Statistics Let the probability of a discrete
event A be P(A). We now introduce two additional probabilities.
The joint probability for events A and B together is written P(
freely extremize S over all its arguments. Thus, for all n we
have S pn = ln pn + 1 + = 0 (1.32) as well as S = X
n pn 1 = 0 . (1.33) From the first of these equations, we obtain
pn = e (1+) , and from the second we obtain X n pn = e (1+)
X n 1 = e(1+) ,
order to invoke the formulae in eqns. 2.27, 2.30, and 2.32, we
need to know the state function E(T, V, N). A landmark
experiment by Joule in the mid19th century established that
the energy of a low density gas is independent of its volume6 .
Essentially,
RT2 V = RT2 ln VB VA (2.68) EA = EB = RT2 1 , (2.69)
hence QAB = EAB + WAB = RT2 ln VB VA . (2.70) BC: This
stage is an adiabatic expansion. We have QBC = 0 (2.71) EBC =
EC EB = R 1 (T1 T2 ) . (2.72) The energy change is
negative, and the heat exchange is
closed path must vanish: I dF = 0 . (2.15) When the cross
derivatives are not identical, i.e. when Ai /xj 6= Aj /xi , the
differential is inexact. In this case, the integral of dF is path
dependent, and does not depend solely on the endpoints. As an
examp
must exist a relation among them. Such relations are known as
equations of state. The most famous example is the ideal gas
law, pV = NkBT , (2.1) 20 CHAPTER 2. THERMODYNAMICS Figure
2.2: The pressure p of a gas is due to an average over space and
time of
S/kB , which is the entropy in units of Boltzmanns constant.
And what we have called E here is really E/kBT , which is energy
in units of Boltzmanns constant times temperature. 3The
function s(x) is the specific entropy. 6 CHAPTER 1. PROBABILITY
1.2.4 Ent
boiling water will do, but anyone who has gone camping in the
mountains knows that water boils at lower temperatures at
high altitude (lower pressure). This phenomenon is reflected in
the phase diagram for H2O, depicted in Fig. 2.4. There are two
special
Xrms are proportional to N1/2 . In the limit N then, the
ratio Xrms/hXi vanishes as N 1/2 . This is a consequence of
the central limit theorem (see 1.4.2 below), and we shall meet
up with it again on several occasions. We can do even better.
We can find t
Bayesian chain of reasoning. If our prior distribution is not
accurate, Bayes theorem will generate incorrect results. One
approach to obtaining the prior probabilities P(Ai ) is to obtain
them from a maximum entropy construction. Chapter 2
Thermodynamics
are (p, V, N), we write dE = E p V,N dp + E V p,N dV + E
N p,V dN . (2.31) Then dQ = E p V,N dp + " E V p,N + p
# dV + " E N p,V # dN . (2.32) 28 CHAPTER 2.
THERMODYNAMICS cp cp cp cp SUBSTANCE (J/mol K) (J/g K)
SUBSTANCE (J/mol K) (J/g K) Air 29.07 1.01
thereof, but it could be any countable set. As an example,
consider the throw of a single sixsided die. Then Pn = 1 6 for
each n cfw_1, . . ., 6. Let An = 0 if n is even and 1 if n is odd.
Then find hAi = 1 2 , i.e. on average half the throws of the die
2.6.4, we will see that how, using an ideal gas as the working
substance of the Carnot engine, this temperature scale
coincides precisely with the ideal gas temperature scale from
2.2.4. 2.6.3 Nothing beats a Carnot engine The Carnot engine
is the most ef