Re: Entropy of early universe

2019-09-16 Thread Lawrence Crowell 
On Saturday, September 14, 2019 at 10:07:28 AM UTC-5, Alan Grayson wrote:
>
>
>
> On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote:
>>
>> If the early universe, say before the emergence of the CMBR, consisted of 
>> a random collection of electrons and photons, wouldn't this correspond to a 
>> *high*, not low entropy? Wouldn't it be analogous to gas with many 
>> possible states? Yet cosmologists seem hard pressed to explain an initial 
>> or early state assuming the entropy is low. AG
>>
>
> Here's an easier question: when Boltzmann defined entropy as S = k * log 
> N, why the log; why not just k*N? AG
>

 Think of the case where you have binary strings of length n. How many 
possible binary string are there with that length? There are N = 2^n. The 
Boltzman log(N) is just the size of the macrostate, where there are 2^n 
possible microstates. This is where the entropy S = kn comes from, for the 
units of Planck area on the horizon count microstates. We have 

S = k ln(N) = k ln(2^n) = k n ln(2).

With the black hole horizon or any horizon this linear chain is replaced by 
a two dimensional table or matrix. The same argument carries over.

LC

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Re: Entropy of early universe

2019-09-16 Thread Alan Grayson 


On Monday, September 16, 2019 at 1:23:09 AM UTC-6, Russell Standish wrote:
>
> On Sat, Sep 14, 2019 at 10:13:27PM -0700, Alan Grayson wrote: 
> > 
> > 
> > On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote: 
> > 
> > If the early universe, say before the emergence of the CMBR, 
> consisted of a 
> > random collection of electrons and photons, wouldn't this correspond 
> to a 
> > high, not low entropy? Wouldn't it be analogous to gas with many 
> possible 
> > states? Yet cosmologists seem hard pressed to explain an initial or 
> early 
> > state assuming the entropy is low. AG 
> > 
> > 
> > When I was an undergraduate I took a course in Classical Thermodynamics 
> and 
> > recall being satisfied that entropy was well-defined. I never took a 
> course in 
> > Classical Statistical Mechanics, but I've seen Boltzmann's equation for 
> S and 
> > wonder how N, the number of possible states is defined. If we have a gas 
> > enclosed in a container, we can divide it into occupation cells of fixed 
> volume 
> > to calcuate S. But why can't we double the number of cells by reducing 
> their 
> > volume by half? How then is S well defined in the case of Classical 
> Statistical 
> > Mechanics? TIA, AG 
>
> It actually isn't. The point bothered me too. The number of states is 
> basically V/h, where V is the volume of phase space occupied by the 
> system, and h a cell size. Therefore, entropy is 
>
> klog V  - klog h 
>
> For a large range of values of h, the second term is just a negligible 
> constant offset to the total entropy. However, as h→0, entropy blows 
> up. And that what classical statistical mechanics tells you. 
>

How could the second term be negligible for large values of h? AG 

>
> Enter quantum mechanics. Heisenberg's uncertainty relation tells us 
> that ΔxΔp ≥ ℏ, so in the above entropy formula, h is constrained to be 
> larger than ℏ³. Quantum mechanics saves classical statistical physics' 
> bacon. Nothing blows up. 
>
> -- 
>
>  
>
> Dr Russell StandishPhone 0425 253119 (mobile) 
> Principal, High Performance Coders 
> Visiting Senior Research Fellowhpc...@hpcoders.com.au 
>  
> Economics, Kingston University http://www.hpcoders.com.au 
>  
>
>

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Re: Entropy of early universe

2019-09-16 Thread Russell Standish 
On Sat, Sep 14, 2019 at 10:13:27PM -0700, Alan Grayson wrote:
> 
> 
> On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote:
> 
> If the early universe, say before the emergence of the CMBR, consisted of 
> a
> random collection of electrons and photons, wouldn't this correspond to a
> high, not low entropy? Wouldn't it be analogous to gas with many possible
> states? Yet cosmologists seem hard pressed to explain an initial or early
> state assuming the entropy is low. AG
> 
> 
> When I was an undergraduate I took a course in Classical Thermodynamics and
> recall being satisfied that entropy was well-defined. I never took a course in
> Classical Statistical Mechanics, but I've seen Boltzmann's equation for S and
> wonder how N, the number of possible states is defined. If we have a gas
> enclosed in a container, we can divide it into occupation cells of fixed 
> volume
> to calcuate S. But why can't we double the number of cells by reducing their
> volume by half? How then is S well defined in the case of Classical 
> Statistical
> Mechanics? TIA, AG

It actually isn't. The point bothered me too. The number of states is
basically V/h, where V is the volume of phase space occupied by the
system, and h a cell size. Therefore, entropy is

klog V  - klog h

For a large range of values of h, the second term is just a negligible
constant offset to the total entropy. However, as h→0, entropy blows
up. And that what classical statistical mechanics tells you.

Enter quantum mechanics. Heisenberg's uncertainty relation tells us
that ΔxΔp ≥ ℏ, so in the above entropy formula, h is constrained to be
larger than ℏ³. Quantum mechanics saves classical statistical physics'
bacon. Nothing blows up.

-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: Entropy of early universe

2019-09-16 Thread John Clark 
On Sat, Sep 14, 2019 at 11:07 AM Alan Grayson 
wrote:

*> Here's an easier question: when Boltzmann defined entropy as S = k * log
> N, why the log; why not just k*N? AG*
>

Because if you define Entropy with a log in there then it is additive for
independent sources; the Entropy of a  coin toss is 1 bit so the Entropy of
10 coin tosses is 10 bits.

John K Clark

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Re: Entropy of early universe

2019-09-15 Thread Lawrence Crowell 
On Sunday, September 15, 2019 at 12:13:27 AM UTC-5, Alan Grayson wrote:
>
>
>
> On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote:
>>
>> If the early universe, say before the emergence of the CMBR, consisted of 
>> a random collection of electrons and photons, wouldn't this correspond to a 
>> *high*, not low entropy? Wouldn't it be analogous to gas with many 
>> possible states? Yet cosmologists seem hard pressed to explain an initial 
>> or early state assuming the entropy is low. AG
>>
>
> When I was an undergraduate I took a course in Classical Thermodynamics 
> and recall being satisfied that entropy was well-defined. I never took a 
> course in Classical Statistical Mechanics, but I've seen Boltzmann's 
> equation for S and wonder how N, the number of possible states is defined. 
> If we have a gas enclosed in a container, we can divide it into occupation 
> cells of fixed volume to calcuate S. But why can't we double the number of 
> cells by reducing their volume by half? How then is S well defined in the 
> case of Classical Statistical Mechanics? TIA, AG
>

There is the classical definition S = ∂E/∂T for isobaric systems. Yet in 
general entropy is a rather subjective and slippery concept. With the 
Boltzmann formula S = k log(Ω) for Ω the volume of phase space any 
uncertainty in Ω results in tiny errors because of the logarithm. 

LC

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Re: Entropy of early universe

2019-09-14 Thread Alan Grayson 


On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote:
>
> If the early universe, say before the emergence of the CMBR, consisted of 
> a random collection of electrons and photons, wouldn't this correspond to a 
> *high*, not low entropy? Wouldn't it be analogous to gas with many 
> possible states? Yet cosmologists seem hard pressed to explain an initial 
> or early state assuming the entropy is low. AG
>

When I was an undergraduate I took a course in Classical Thermodynamics and 
recall being satisfied that entropy was well-defined. I never took a course 
in Classical Statistical Mechanics, but I've seen Boltzmann's equation for 
S and wonder how N, the number of possible states is defined. If we have a 
gas enclosed in a container, we can divide it into occupation cells of 
fixed volume to calcuate S. But why can't we double the number of cells by 
reducing their volume by half? How then is S well defined in the case of 
Classical Statistical Mechanics? TIA, AG

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Re: Entropy of early universe

2019-09-14 Thread Jason Resch 
On Sat, Sep 14, 2019, 1:35 PM Jason Resch  wrote:

>
>
> On Sat, Sep 14, 2019, 10:07 AM Alan Grayson 
> wrote:
>
>>
>>
>> On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote:
>>>
>>> If the early universe, say before the emergence of the CMBR, consisted
>>> of a random collection of electrons and photons, wouldn't this correspond
>>> to a *high*, not low entropy? Wouldn't it be analogous to gas with many
>>> possible states? Yet cosmologists seem hard pressed to explain an initial
>>> or early state assuming the entropy is low. AG
>>>
>>
>> Here's an easier question: when Boltzmann defined entropy as S = k * log
>> N, why the log; why not just k*N? AG
>>
>
>
> I don't know the relationship between heat and information, I think it is
> relevant to the Bekenstein bound and black hole information, and also the
> Landauer limit, but there's another definition of entropy in information
> theory: https://en.m.wikipedia.org/wiki/Entropy_(information_theory)
>
> The information theoretical definition of entropy is measured in bits
> (binary digits).  The reason for the logarithm is it takes Log2(N) bits to
> represent N states.  There's nothing special about the base you can also
> say it takes Log10(N) decimal digits to encode a number N.
>
> Jason
>


I found this article which adds more detail:

https://en.m.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory

Jason

>

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Re: Entropy of early universe

2019-09-14 Thread Jason Resch 
On Sat, Sep 14, 2019, 10:07 AM Alan Grayson  wrote:

>
>
> On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote:
>>
>> If the early universe, say before the emergence of the CMBR, consisted of
>> a random collection of electrons and photons, wouldn't this correspond to a
>> *high*, not low entropy? Wouldn't it be analogous to gas with many
>> possible states? Yet cosmologists seem hard pressed to explain an initial
>> or early state assuming the entropy is low. AG
>>
>
> Here's an easier question: when Boltzmann defined entropy as S = k * log
> N, why the log; why not just k*N? AG
>


I don't know the relationship between heat and information, I think it is
relevant to the Bekenstein bound and black hole information, and also the
Landauer limit, but there's another definition of entropy in information
theory: https://en.m.wikipedia.org/wiki/Entropy_(information_theory)

The information theoretical definition of entropy is measured in bits
(binary digits).  The reason for the logarithm is it takes Log2(N) bits to
represent N states.  There's nothing special about the base you can also
say it takes Log10(N) decimal digits to encode a number N.

Jason

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Re: Entropy of early universe

2019-09-14 Thread Lawrence Crowell 
On Saturday, September 14, 2019 at 8:12:34 AM UTC-5, Alan Grayson wrote:
>
> If the early universe, say before the emergence of the CMBR, consisted of 
> a random collection of electrons and photons, wouldn't this correspond to a 
> *high*, not low entropy? Wouldn't it be analogous to gas with many 
> possible states? Yet cosmologists seem hard pressed to explain an initial 
> or early state assuming the entropy is low. AG
>

The inflationary manifold has a cosmological constant Λ ≈ (1/cℓ_p)^2, where 
c is a number c < 1 and with a value of around 10^{-3}  This means Λ ≈ 
10^{64}m^{-2}, which is to be compared to today's value of Λ ≈ 
10^{-52}m^{-2}. What is still studied is the process by which this false 
vacuum, or false cosmological constant, transitioned to the value today. If 
we take the scenario that inflation started at 10^{-36} sec this would mean 
there was a transition of sorts from Λ ≈ 10^{64}m^{-2} → 10^{56}m^{-2} and 
in one scenario this transitions in a slow role to a lower value and 
transitions again to the low value known. 

The amount of information in a region bounded by a cosmological horizon 
with area A is S ≈  kA/( cℓ_p)^2 = k/c^2 by the Bousso bound. This means a 
tiny region bounded by this cosmological horizon only about 10^{-32}m 
across had a total entropy of S ≈ k×10^{6} for k = 1.38×10^{-23}j/K. With 
this transition this was some 8 orders of magnitude larger. So as a result 
the entropy of the earliest universe was quite small.

LC

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Re: Entropy of early universe

2019-09-14 Thread Alan Grayson 


On Saturday, September 14, 2019 at 7:12:34 AM UTC-6, Alan Grayson wrote:
>
> If the early universe, say before the emergence of the CMBR, consisted of 
> a random collection of electrons and photons, wouldn't this correspond to a 
> *high*, not low entropy? Wouldn't it be analogous to gas with many 
> possible states? Yet cosmologists seem hard pressed to explain an initial 
> or early state assuming the entropy is low. AG
>

Here's an easier question: when Boltzmann defined entropy as S = k * log N, 
why the log; why not just k*N? AG

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Re: Entropy of early universe

2019-09-14 Thread Jason Resch 
Lazer says the expansion of the universe creates an increased difference
between the current entropy and the maximum possible entropy:
https://www.informationphilosopher.com/solutions/scientists/layzer/growth_of_order/

Thereby introducing room for entropy to increase further, and giving the
appearance of low entropy initial conditions.

Jason

On Sat, Sep 14, 2019, 8:12 AM Alan Grayson  wrote:

> If the early universe, say before the emergence of the CMBR, consisted of
> a random collection of electrons and photons, wouldn't this correspond to a
> *high*, not low entropy? Wouldn't it be analogous to gas with many
> possible states? Yet cosmologists seem hard pressed to explain an initial
> or early state assuming the entropy is low. AG
>
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> 
> .
>

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Entropy of early universe

2019-09-14 Thread Alan Grayson 
If the early universe, say before the emergence of the CMBR, consisted of a 
random collection of electrons and photons, wouldn't this correspond to a 
*high*, not low entropy? Wouldn't it be analogous to gas with many possible 
states? Yet cosmologists seem hard pressed to explain an initial or early 
state assuming the entropy is low. AG

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