# Nisan

Karma: 6,264

• Scaling temperature up by a factor of 4 scales up all the velocities by a factor of 2 [...] slowing down the playback of a video has the effect of increasing the time between collisions [....]

Oh, good point! But hm, scaling up temperature by 4x should increase velocities by 2x and energy transfer per collision by 4x. And it should increase the rate of collisions per time by 2x. So the rate of energy transfer per time should increase 8x. But that violates Newton’s law as well. What am I missing here?

• constant volume

Ah, so I’m working at a level of generality that applies to all sorts of dynamical systems, including ones with no well-defined volume. As long as there’s a conserved quantity , we can define the entropy as the log of the number of states with that value of . This is a univariate function of , and temperature can be defined as the multiplicative inverse of the derivative .

if the proportionality depends on thermodynamic variables

By

I mean

for some constant that doesn’t vary with time. So it’s incompatible with Newton’s law.

This asymmetry in the temperature dependence would predict that one subsystem will heat faster than the other subsystem cools

Oh, the asymmetric formula relies on the assumption I made that subsystem 2 is so much bigger than subsystem 1 that its temperature doesn’t change appreciably during the cooling process. I wasn’t clear about that, sorry.

• 17 Jan 2024 8:28 UTC
2 points
0

Yeah, as Shankar says, this is only for conduction (and maybe convection?). The assumption about transition probabilities is abstractly saying there’s a lot of contact between the subsystems. If two objects contact each other in a small surface area, this post doesn’t apply and you’ll need to model the heat flow with the heat equation. I suppose radiative cooling acts abstractly like a narrow contact region, only allowing photons through.

• 17 Jan 2024 8:17 UTC
4 points
0

I am suspicious of this “Lambert’s law”. Suppose the environment is at absolute zero—nothing is moving at all. Then “Lambert’s law” says that the rate of cooling should be infinite: our object should itself instantly drop to absolute zero once placed in an absolute-zero environment. Can that be right?

We’re assuming the environment carries away excess heat instantly. In practice the immediate environment will warm up a bit and the cooling rate will become finite right away.

But in the ideal case, yeah, I think instant cooling makes sense. The environment’s coldness is infinite!

• 17 Jan 2024 8:07 UTC
4 points
0

Oh neat! Very interesting. I believe your argument is correct for head-on collisions. What about glancing blows, though?

Assume two rigid, spherical particles with the same mass and radius.

Pick a coordinate system (at rest) where the collision normal vector is aligned with the x-axis.

Then move the coordinate system along the x axis so that the particles have equal and opposite x-velocities. (The y-velocities will be whatever.) In this frame, the elastic collision will negate the x-velocities and leave the y-velocities untouched.

Back in the rest frame, this means that the collision swaps the x-velocities and keeps the y-velocities the same. Thus the energy transfer is half the difference of the squared x-velocities, .

I’m not sure that’s proportional to ? The square of the x-velocity does increase with temperature, but I’m not sure it’s linear. If there’s a big temperature difference, the collisions are ~uniformly distributed on the cold particle’s surface, but not on the hot particle’s surface.

# New­ton’s law of cool­ing from first principles

16 Jan 2024 4:21 UTC
23 points
• 8 Jan 2024 1:41 UTC
2 points

I’d love if anyone can point me to anywhere this cooling law (proportional to the difference of coldnesses) has been written up.

Also my assumptions about the dynamical system are kinda ad hoc. I’d like to know assumptions I ought to be using.

• We can derive Newton’s law of cooling from first principles.

Consider an ergodic discrete-time dynamical system and group the microstates into macrostates according to some observable variable . ( might be the temperature of a subsystem.)

Let’s assume that if , then in the next timestep can be one of the values , , or .

Let’s make the further assumption that the transition probabilities for these three possibilities have the same ratio as the number of microstates.

Then it turns out that the rate of change over time is proportional to , where is the entropy, which is the logarithm of the number of microstates.

Now suppose our system consists of two interacting subsystems with energies and . Total energy is conserved. How fast will energy flow from one system to the other? By the above lemma, is proportional to .

Here and are the coldnesses of the subsystems. Coldness is the inverse of temperature, and is more fundamental than temperature.

Note that Newton’s law of cooling says that the rate of heat transfer is proportional to . For a narrow temperature range this will approximate our result.

• Wow, that’s a lot of kale. Do you eat 500g every day? And 500g is the mass of the cooked, strained kale?

• I wonder why Gemini used RLHF instead of Direct Preference Optimization (DPO). DPO was written up 6 months ago; it’s simpler and apparently more compute-efficient than RLHF.

• Is the Gemini org structure so sclerotic that it couldn’t switch to a more efficient training algorithm partway through a project?

• Is DPO inferior to RLHF in some way? Lower quality, less efficient, more sensitive to hyperparameters?

• Maybe they did use DPO, even though they claimed it was RLHF in their technical report?

• Another example is the obfuscated arguments problem. As a toy example:

For every cubic centimeter in Texas, your missing earring is not in the cubic centimeter.

Therefore, your missing earring is not in Texas.

Even if the conclusion of the argument is a lie, each premise is spot-checkable and most likely true. The lie has been split up into many statements each of which is only slightly a lie.

• 21 Oct 2023 19:44 UTC
LW: 3 AF: 1
AF
in reply to: Cleo Nardo’s comment

Thanks! For convex sets of distributions: If you weaken the definition of fixed point to , then the set has a least element which really is a least fixed point.

• Hyperbolic growth

The differential equation , for positive and , has solution

(after changing the units). The Roodman report argues that our economy follows this hyperbolic growth trend, rather than an exponential one.

While exponential growth has a single parameter — the growth rate or interest rate — hyperbolic growth has two parameters: is the time until singularity, and is the “hardness” of the takeoff.

A value of close to zero gives a “soft” takeoff where the derivative gets high well in advance of the singularity. A large value of gives a “hard” takeoff, where explosive growth comes all at once right at the singularity. (Paul Christiano calls these “slow” and “fast” takeoff.)

Paul defines “slow takeoff” as “There will be a complete 4 year interval in which world output doubles, before the first 1 year interval in which world output doubles.” This corresponds to . (At , the first four-year doubling starts at and the first one-year doubling starts at years before the singularity.)

So the simple hyperbola with counts as “slow takeoff”. (This is the “naive model” mentioned in footnote 31 of Intelligence Explosion Microeconomics.)

Roodman’s estimates of historical are closer to (see Table 3).

• 25 Sep 2023 22:13 UTC
2 points
0

Ah, beginning-of-line-text is nice. It skips over the initial # or //​ of comments and the initial * of Org headings. I’ve now bound it to M-m.

• Consider seeing a doctor about the panicky and stressed feelings. They may test you for hormone imbalances or prescribe you antianxiety medication.

• 6 Jul 2023 19:29 UTC
2 points
0
in reply to: Wei Dai’s comment

A long reflection requires new institutions, and creating new institutions requires individual agency. Right? I have trouble imagining a long reflection actually happening in a world with the individual agency level dialed down.

A separate point that’s perhaps in line with your thinking: I feel better about cultivating agency in people who are intelligent and wise rather than people who are not. When I was working on agency-cultivating projects, we targeted those kinds of people.

• What’s more, even selfish agents with de dicto identical utility functions can trade: If I have two right shoes and you have two left shoes, we’d trade one shoe for another because of decreasing marginal utility.