Sunday 5 June 2022

A research programme after Leibniz for a future physics

In "Einstein’s Unfinished Revolution", Penguin Books, 2019, Lee Smolin introduces principles through which to develop fundamental physics. So, they are not mathematical or logical principles but founding elements for thinking about and then formulating physical theories.

Introduce five closely related principles for a future physics:

  1. The principle of background independence 
  2. The principle that space and time are relational 
  3. The principle of causal completeness 
  4. The principle of reciprocity 
  5. The principle of the identity of indiscernibles.
These are all aspects, claims Smolin, of what Leibniz called the principle of sufficient reason (PSR). he interprets the principle as follows: 
Every time we identify some aspect of the universe which seemingly might be different, we will discover, on further examination, a rational reason why it is so and not otherwise.
 This is not quite standard but then, if we even go back to Leibniz, we have a number of versions. We shall examine the formulation and the five principles in light of their origin in the philosophy of Leibniz and then go on to discuss them as principles for developing physics especially in dealing with quantum mechanics and theories of space and time.

The promotion of the principle by Leibniz

Although the PSR is most closely associated with Leibniz there is an earlier formulation by Spinoza
Nothing exists of which it cannot be asked, what is the cause (or reason) [causa (sive ratio)], why it exists.
The version given by Smolin has an epistemic flavour while that of Spinoza is ontological. As Smolin is defending a realist view of physics an ontological formulation may be preferred. Leibniz provided a mixture of logical, ontological and epistemic formulations. The modern recovery of the reputation of Leibniz is based on his achievements as a logician and here is the version from the Monadology (not Leibniz's title)
31. Our reasonings are based on two great principles, that of contradiction, in virtue of which we judge that which involves a contradiction to be false, and that which is opposed or contradictory to the false to be true.
32. And that of sufficient reason, by virtue of which we consider that we can find no true or existent fact, no true assertion, without there being a sufficient reason why it is thus and not otherwise, although most of the time these reasons cannot be known to us. 

Saturday 4 June 2022

Ergodicity and investment growth

The status of Ergodicity Economics as a minority topic in mathematical economics may be about to change due to the profile it has achieved through a special 'perspective' article published in Nature in December 2019 authored by Ole Peters of the London Mathematical Laboratory.

The article itself provides a clear introduction that presents sufficient mathematics without being overly pedantic. The main point has been well known and put on a rigorous foundation in mathematical physics for several decades. The point being that the time average of a dynamical variable is not equal to the static expectation value over the associated probability distribution over that variable. The claim that this ignored in main stream economics seems incredible given the repeated claims that mathematics has too tight a hold on the subject currently. However the effects that Peters claims does seem to stand up to scrutiny.

The model that is used to illustrate is a simple gamble where with equal probability you can increase your pot (wealth) $W$ by $A$% or lose $B$%. If $A > B$ then the expected outcome is positive and traditionally the gamble is considered acceptable. In general the outcome is
$$<\Delta W> = W \frac{A - B}{2 \times 100} .$$
This simple game of chance could also be thought of as the outcome of a risky investment or a purchase that has associated unknowns.

However when the stakes are high most people are disinclined to accept the such a bet. High stake would mean something like betting your house. A bad outcome would leave you with significantly reduced  total wealth. The rejection of the gamble is often declared irrational as  you are rejecting an expected win.  But something slippery is going on; "expected" is being used in two distinct ways. The ordinary language use means what is likely happen and what is very likely to happen is that either your wealth increase significantly or it decreases significantly. The technical meaning is that it is the average over the probability distribution; Equation for $\Delta W$, above is that average for the gamble discussed.

Now one way to day with a large one off loss is to spread your bets over time. That is iterate the game. So what is the expected win if the game is played $N$ times? To address this we will have to introduce a more formal mathematical model. Let $s(0)$ be the initial stake and let the stake to iteration $n$ be
$$s(n) = \prod_{i=1}^{n} r_i s(0) $$
where $r_i = a$ with probability a half and $b$ with probability a half, where
$$a = 1 - A/100$$

$$b = 1 + B/100$$.
A further technical assumption is that each gamble is independent of the previous one, which gives for the expected stake:
$$<s(n)> = \prod_{i=1}^{n} <r_i >s(0) $$
$$<s(n)> = <r_i>^n s(0)= <r>^n s(0) $$
$$<r> =\frac{a+b}{2}$$
where we have used the fact that each gamble has the same probability distribution.
As long as $<r>$ is greater than one the stake will grow. Is the calculated expectation value what is to be expected? To test this we will look at what happens over time to $s(n)$, that is as $n$ increases. For large $n$, noting that the order in which $a$ or $b$ is randomly selected does not affect the value of $s(n)$. Therefore
$$s(n) = a^{m} b^{n-m} s(0) $$
\[\lim_{n \to \infty} s(n)^{1/n} = \hat{s} =\sqrt{ a b} s(0)\].

So this increases exponentially with rate $\sqrt{ a b}$ if $ab > 1$. Let us first consider the example with parameters used by Peters. The result for $a=0.6$ and $b=1.5$ is shown below for $100$ histories. This case shows that for these parameters playing the game leads to almost certain loss, amounting to ruin or almost complete loss of stake, but the expectation value estimate predicts an exponentially growing gain. For these parameters $\sqrt{ab} < 1$ so the analysis has told us to anticipate this. Here anticipation and expectation are not the same.

Lets keep $a=0.6 but now let $ab$ be greater than 1. This gives

 This give most histories growing, just, after $25000$ iterations. As  $\hat{r} =1.0003$ and now $,r. = 1.366$. By making   $<r>$ slightly bigger we get into a situation where the game becomes almost always favourable to the player.

A more realistic gamble such as investment in carefully chosen stock it likely to vary only a few percent between iterations. This is the situation shown below.

Here when $a$ and $b$ are close there is less uncertainty and the expectation value is not such a terrible estimate as in the previous examples. Perhaps the positive return is unrealistically high. An iterated sample with a 4% loss and 5% gain is shown below.

Tuesday 15 March 2022

Energy Science and Technology priorities to achieve "net-zero" GHG emissions

I am no expert on energy and the environment, having spent the latter half of my career in information systems engineering; especially security, safety and automation. However, I have also worked as a trouble shooting systems generalist and it is this experience that I want to try to bring to the energy challenge in tackling climate change.

Grangemouth refinery
The government is committing to  “Net Zero” greenhouse gas (GHG) emissions by 2050. This is good news but the means of achieving it are critical. To tackle climate change innovation is still urgently needed and it must come quickly because implementation time scales for new technology in complex systems are so slow. Often implementation to operation requires more than a decade to get on-line, as can be seen from the proposed Small Modular Reactor power plant that will not see operation before 2030. However,  even at this stage, given that the best scientific opinion has been clear on climate change for decades, there is still no consensus on the mix of energy solutions required and priorities. There is a continuing environmental movement opposition in principle to nuclear energy, despite its zero GHG emissions when in operation. Ensuring that the identified priorities are the right ones means recognising those environmental and societal factors that contribute to global warming and disentangling them from the real but different concerns on air quality and pollution of oceans and waterways.  The concerns about species extinction would also benefit from the clarity afforded by distinguishing between what is and what is not a due to GHG emissions.

Rolls-Royce consortium concept Small Modular Reactor facility
We need to unblock the system to act with urgency but remain focused on a plan. All this requires an increase in tempo. Suggestions on how to do this have been provided in a report prepared for the Aldersgate Group: Accelerating innovation towards net zero emissions. They identify six (not five) key actions for government policy to accelerate low carbon innovation in the UK :
  1. Increase ambition in demonstrating complex and high capital cost technologies
  2. Create new markets to catalyse early deployment and move towards widespread commercialisation
  3. Use concurrent innovations, such as digital technologies, to improve system efficiency and make new products more accessible and attractive to customers.
  4. Use existing or new institutions to accelerate critical innovation areas and co-ordinate early stage deployment.
  5. Harness trusted voices to build consumer acceptance.
  6. Align innovation policy in such a way that it strengthens the UK’s industrial advantages and increases knowledge spill overs between businesses and sectors
The report says that implementing these lessons will require a further increase in government support for innovation – through both research, development and demonstration and through deployment policies to create new markets. Government doe snot have good record in implementing complex infrastructure projects. As well as policies to create new markets, regulation and market correction should be implemented to channel the the infrastructure and engineering capabilities of the the oil and gas, as well as the defence and petro-chemical industries. As has been argued persuasively by the Nobel Prize winning economist William Nordhaus, many problems should be solved by robust carbon pricing, preferably through a carbon tax. This will release current market potential and create new market opportunities as well as generating funding for innovation.

If cost was no obstacle, land and material resources available, then current renewable energy technologies and storage methods would be sufficient. But this is not the case; there are trade-offs with costs and benefits of any course of action need to be weighed. There are many challenges and perhaps the greatest are political in both needing to convince the populations of almost all countries in the world to sacrifice their current quality of life to mitigate a predicted greater cost but one that will impact grand children and finding ways to ensure that the free loader effect does no disrupt the good intentions achieved through the Paris protocol. However if this political challenges are achieved further innovation will still be required and if the political challenges are not addressed or only partially then innovation become the only hope. However, the International Energy Agency (IEA) identifies a bottleneck in innovation. The investment in R&D for low carbon technology is not growing. IEA are tracking key technologies and of a set of 39 only 7 are on track to meet Paris targets, 20 need remedial action and 13 are off track. Those that are on track are: solar photovoltaics (PV)bioenergy for powerenergy storageelectric vehicles (EVs)raillighting  and data centres. It must be emphasised that these are on track, not complete, further investment and maintained efforts are required. To concentrate on the power domain the following technologies are tracked by the IEA:

Now, not all are equally critical but there are systemic dependencies so that the full benefit of net-zero carbon energy generation can be delivered through efficient power transmission with capacity for storage. Obviously the problem with coal fired power is that we are still using too much of it and this can only be solved by replacement technologies. For wind the key problem is not primarily technical, although there are improvements to be made, but regulation, planning and consultation. In those areas that are difficult to de-carbonise there will need to be compensating carbon capture technology. In nuclear technical work to increase modularisation and scalability is required as well safety and security by design. The major objections to nuclear of waste, safety, security and cost are being addressed but efforts must continue.

We need a short to medium term boost in funding to achieve the needed acceleration. This could be provided as a dividend from imposing a carbon tax or other robust form of carbon pricing. As well as providing revenue for R&D it also provides a steering mechanism because the tax will help correct for the failure of the energy market and provide path to exploitation for the innovations. Speed of exploitation remains a challenge with a quagmire of regulations, permissions and consultations to wade through in addition to the technical challenges that are always present in scaling up form proof of concept to operational system. Eventually an emergency situation will require emergency powers but climate change is a slow motion emergency both in the climatic development in response to increased green house gas concentration and in the climates response to mitigation. In the end it will come down to whether there is the social and political will to carry through an emergency response.