A phase-space description of the Earth System in the Anthropocene

Abstract

Based on a dynamic systems approach to the Landau-Ginzburg model, a phase space description of the Earth System (ES) in the transition to the Anthropocene is presented. It is shown that, for a finite amount of human-driven change, there is a stable equilibrium state that is an attractor of trajectories in the system's phase space and corresponds to a Hothouse Earth scenario. Using the interaction between the components of the ES, it is argued that, through the action of the Technosphere, mitigation strategies might arise for which the deviation of the ES temperature from the Holocene average temperature is smaller.

Full text

A phase-space description of the Earth System in the Anthropocene
O. Bertolami, F. Francisco
Departamento de Física e Astronomia and Centro de Física do Porto, Faculdade de Ciências, Universidade do Porto, Rua do
Campo Alegre 687, 4169-007 Porto, Portugal
Abstract
Based on a dynamic systems approach to the Landau-Ginzburg model, a phase space description of the Earth
System (ES) in the transition to the Anthropocene is presented. It is shown that, for a finite amount of
human-driven change, there is a stable equilibrium state that is an attractor of trajectories in the system’s
phase space and corresponds to a Hothouse Earth scenario. Using the interaction between the components of
the ES, it is argued that, through the action of the Technosphere, mitigation strategies might arise for which
the deviation of the ES temperature from the Holocene average temperature is smaller.
Keywords: Anthropocene, Earth System, Phase Transition
1. Introduction
Recently, we have proposed that the Earth System
(ES) transition from the Holocene to other stable ge-
ological eras was similar to a phase transition and
could be described by the Landau-Ginzburg theory
(LGT) [1]. This description suggests that the rele-
vant thermodynamic variable to specify the state of
the ES is the free energy, F, and that a relevant or-
der parameter, ψ, is the relative temperature devi-
ation from the Holocene average temperature, hTHi,
ψ= (T− hTHi)/hTHi.
This physical framework allows for establishing the
state of the ES in terms of the relevant physical vari-
ables, (η, H ), where ηcorresponds to the astronom-
ical, geophysical and internal dynamical causes of
change, while Hstands for the human activities.
The Landau-Ginzburg model allows for obtaining
the so-called Anthropocene equation, i.e., the evolu-
tion equation of the ES once it is dominated by the
human activities, and also to show that the transition
from the Holocene conditions to the Anthropocene are
indeed associated to a great acceleration of the human
activities. This is consistent with the empirical obser-
vation that the Anthropocene started by the second
half of the 20th century [2].
Email addresses: orfeu.bertolami@fc.up.pt (O.
Bertolami), frederico.francisco@fc.up.pt (F. Francisco)
In the present work we perform a phase space anal-
ysis of the temperature field, ψ, and show that the
recently discussed Hothouse Earth scenario [3] corre-
sponds, for a finite amount of human driven change,
to a stable minimum and, therefore, an attractor of
the trajectories of the dynamical system that de-
scribes the ES in terms of the Landau-Ginzburg
model. We also discuss how to “engineer” other pos-
sible minima for the ES.
This paper is organized as follows: in the next sec-
tion we review the Landau-Ginzburg model proposal,
discuss the Anthropocene Equation (AE), the dynam-
ical system emerging from this description and its
phase space. In section 3, the Hothouse earth state is
shown to be an attractor of trajectories in the phase
space of the ES and we discuss how to engineer new
minima using new terms in the free energy. We also
discuss the interactions between different terms of the
planetary boundaries [4]. This reinforces the impor-
tance of maintaining the ES within the so called safe
operating space [5, 6]. Finally, in section 4 we present
our conclusions.
2. The Anthropocene Equation phase-space
In general terms, a dynamical system is any sys-
tem that evolves in time. It is mathematically de-
scribed by an ordinary differential equation (ODE) of
the form ˙x=f(x, t), where ˙x=dx/dt,tis the time
Preprint submitted to November 15, 2018
arXiv:1811.05543v1 [physics.ao-ph] 9 Nov 2018

and xrepresents a state of the system. The phase
space (x, ˙x), for a dynamical system, is fully speci-
fied by the space of coordinates, x(t), which hence
contains all possible states of the system. An impor-
tant class of dynamical systems, called autonomous,
do not explicitly depend on time and their evolution
equation has the form ˙x=f(x). A dynamical system
can also depend on parameters, α, with an evolution
equation of the form ˙x=f(x, α).
For a given set of initial conditions, corresponding
to a state x0in the phase space, we can establish
and solve the initial value problem with the evolution
equation. Its solution is a function x(t)describing
the trajectory or orbit of the dynamical system in the
phase space.
The study of dynamical systems reveals many of its
properties, notably the existence of attractors, with-
out the need for solving explicitly all relevant initial
value problems.
Dynamical systems in physics are described either
through a Lagrangian function and the least action
principle or the equivalent Hamiltonian formulation,
whether it is in classical, statistical or quantum prob-
lems.
The dynamics of the system is encapsulated in a
Lagrangian function, L(q, ˙q, t), of a set of general-
ized coordinates of the system, q, their time deriva-
tives, ˙qand time itself, t. The time integral of the
Lagrangian is a quantity called action and the min-
imization of the action amounts to determining the
evolution equations of the system, the so-called Euler-
Lagrange equations. Equivalently, we can derive a
Hamiltonian function, H(q, p, t), where pis the canon-
ical conjugate momentum, which is defined below.
In conservative physical applications, H(q, p, t)rep-
resents the total energy of the system, and lead to
evolution equations equivalent to the Euler-Lagrange
equations.
In Ref. [1], we have introduced the free energy of
the ES near a phase transition as described by the
LGT:
F(η, H ) = F0+a(η)ψ2+b(η)ψ4−h(η)Hψ. (1)
Despite its phenomenological nature, this description
of the free energy, which is a thermodynamic poten-
tial, fits quite well in the Hamiltonian formalism to
describe the dynamics of the ES.
The Hamiltonian system is given in terms of a set
of canonical coordinates and conjugate canonical mo-
menta. The Hamiltonian function is defined as
H(q, p) = p˙q− L(q, ˙q),(2)
where p=∂L/∂ ˙qis the canonical conjugate momen-
tum to the canonical coordinate qand L(q. ˙q)is the
ES Lagrangian. We have already omitted the time
dependence of the Lagrangian since we have assumed
that the free energy does not depend explicitly on
time.
The most general set of canonical coordinates for
the ES in this description should include, not only the
order parameter ψ, but also the natural and human
drivers, ηand H, respectively, thus q= (ψ, η, H ).
In addition to the potential which we have al-
ready identified with the free energy, the Lagrangian
should include a set of kinetic terms for the canon-
ical coordinates. The simplest possible kinetic term
is a quadratic term proportional to the squared first
derivative of each coordinate. This way, we write the
Lagrangian as
L(q, ˙q) = µ
2˙
ψ2+ν
2˙η2−F0
−a(η)ψ2−b(η)ψ4+h(η)Hψ, (3)
where µand νare constants.
We have argued that during the Anthropocene the
ES is dominated by the effects of human activities. In
terms of a dynamical systems description, this means
that these have a much larger and faster effect than
the longer time scales of natural factors. For this rea-
son, for a study of the ES centred in the last century
and currently we can safely drop the term in ˙η2. How-
ever, any analysis of the ES for geological time-scales
would have to include this term as well.
Notice that we could have introduced a kinetic term
for the human activities, ˙
H2. This term could de-
scribe how the ES at large affects the human activi-
ties themselves. Although this feedback loop exists,
we shall assume instead that His an external force.
We shall start with a simplified analysis focusing
only on the order parameter used in the LGT formu-
lation, ψ, representing temperature, undoubtedly the
key thermodynamic and climatic state variable. The
additional terms may become relevant as more encom-
passing descriptions of the ES arise. Our Lagrangian,
simplified along this reasoning, becomes
L(ψ, ˙
ψ) = µ
2˙
ψ2−aψ2−bψ4+hHψ, (4)
where a,band hare constants (cf. Ref. [1]) and we
have dropped the constant F0since it will not affect
the dynamics of the system.
2

Figure 1: Stability landscape of the ES in terms of ψand H.
The canonical momentum can be then obtained as
p=∂L
∂˙
ψ=µ˙
ψ. (5)
We now have all the ingredients to write the Hamil-
tonian of the ES
H(ψ, p) = p2
2µ+aψ2+bψ4−hHψ. (6)
Finally, we can use Hamilton’s equations
˙
ψ=∂H
∂p ,˙p=−∂H
∂ψ .(7)
to obtain the evolution equation of the dynamical sys-
tem in the phase space (ψ, ˙
ψ):
˙
ψ=p
µ,˙p=−2aψ −4bψ3+hH. (8)
With these equations, we can plot the phase space
portrait of the dynamical system, allowing us to
graphically identify the orbits and the attractors. To
start with, we should have in mind the stability land-
scape of the temperature field, ψ, depicted in Fig-
ure 1. In the center of the valley for H= 0 we have
the Holocene minimum described in Ref. [1]. Moving
away from H= 0, we clearly see that the human
intervention opens up a deeper, and therefore more
stable, hotter minimum that we can identify with the
Hothouse Earth [3].
We can now examine the possible trajectories given
the dynamical system, Eq. (8). As mentioned, in this
description of the system, ηis fixed and His treated
as a parameter with a temporal dependence. The ES
dynamics is reflected in the phase space and, thus, on
the position and strength of its attractors, as exem-
plified in Figure 2 for increasing values of H.
We can obtain the dynamical system orbits analyt-
ically for a simpler case, but which is still useful to to
acquire some understanding about the phase space.
H=0
ψ
p
H1>0
ψ
p
H1>H2ψ
p
Figure 2: Phase portrait examples of the ES dynamical system
as obtained from Eq. (8)
3

ψ
ψ

Figure 3: Trajectory of the ES in the phase space for H(t) = t
and initial conditions (ψ, p)(0) = (0,0), modelling the depar-
ture from the Holocene equilibrium.
If b'0, we can drop the cubic term in Eq. (8), the
equations of motion now correspond to a harmonic
oscillator with an external force H(t).
We further consider a specific function of time for
the external force, in this case H(t) = H0t. Our sim-
plified system now has equations of motion of the form
µ¨
ψ=−2aψ +hH0t. (9)
If we assume a departure from equilibrium, then
˙
ψ(0) = 0 and the solution is given by
ψ(t) = ψ0cos(ωt) + αt, (10)
where ω=p2a/µ is an angular frequency, α=
hH0/2aand ψ0is an arbitrary constant fixed by the
initial conditions of the problem.
It can be shown that this solution corresponds to
elliptical trajectories in the phase space with moving
foci of the form
Ψ2
ψ2
0
+˙
Ψ2
ψ2
0ω2= 1,(11)
where Ψ = ψ+αt.
These results provide a qualitative picture that still
holds after the cubic term is reintroduced. Its effect is
to slightly deform the ellipse and to slow its movement
to towards higher values of ψ. When we numerically
solve the equations of motion, Eq. (8), we can then
plot the trajectory of the dynamical system for any
given initial conditions. An example is shown in Fig-
ure 3. Notice that our analysis could be generalized
for the case where H(t)has a quadratic, cubic or,
indeed, any time dependence.
Still, our model so far presents the picture of an
ES perpetually orbiting around the equilibrium point.
ψ
p
Figure 4: Phase portrait example of the Earth dynamical sys-
tem with a Rayleigh dissipative term as obtained from Eq.(13).
Compare with the central panel in Figure2.
This is due to the fact that our description up to this
point is fully conservative.
All real physical systems have some dissipative
character that ensures the system always moves to-
wards and eventually reaches a stable equilibrium. In
thermodynamics, this is equivalent to an increase in
the entropy, S, which we know should always grow
with time. Our description of the ES is based on its
free energy, F=U−T S, where Uis internal energy
and Tis the thermodynamic temperature. If the in-
ternal energy remains constant, then the second law
of thermodynamics ensures that the system will move
towards its minimum attainable free energy.
The simplest way to insert a dissipative term in
the Lagrangian formalism is through a Rayleigh func-
tion, R=1
2k˙
ψ2, where kis a positive constant, which
gives a contribution to the equation of motion ∂R/∂ ˙
ψ.
With this additional term, Eq. (8) becomes
µ¨
ψ=−2aψ −4bψ3+hH −k˙
ψ, (12)
or, in terms of the momentum p,
˙p=−2aψ −4bψ3+hH −k
µp. (13)
Notice that Eq. (12) (or Eq. (13)) could itself be re-
gard as an Anthropocene equation, as its solutions
allow for predicting the temperature field for a given
evolution of the human forcing. However, in the liter-
ature, the designation Anthropocene equation is re-
served for the evolution equation of the ES itself,
which was obtained in Ref.[1] through the LGT.
4

The phase portrait of the dynamical system with a
dissipative term shows the orbits changing from de-
formed ellipses to spirals towards the free energy min-
imum, as depicted in Figure 4.
The existence of a stable equilibrium points in dy-
namical systems can be shown through the Lyapunov
theorem.
A critical point (ψc, pc)is said to be Lyapunov sta-
ble if any trajectory starting with a given neighbour-
hood of that point remains within a finite neighbour-
hood of it. The orbits of the ES in Figure 2 for each
constant value of Hare Lyapunov stable around the
minimum of the free energy.
If, additionally, there exists a finite neighbourhood
of the critical point within which all trajectories con-
verge to the critical point, then that critical point is
said to be asymptotically stable. When a dissipative
term is added to the ES evolution equation with con-
stant H, the critical points in the minimum of the free
energy becomes asymptotically stable. In this case,
the critical point is also an attractor of trajectories.
In the scenario where Hdepends on time, this
will cause a continuous shift of the critical point that
causes the kind of trajectories exemplified in Figures 3
and 5, respectively, with and without a dissipative
term. Strictly speaking, the system is only stable
if H(t)is bounded. The Lyapunov condition is en-
sured for our system as, from Eq. (13), we can see
that p(0) ≤hH −(k/µ)p(0) for ψ > 0, implying that
there is a state (ψc, pc)for which pc= 0, for a finite
H. In that case, the ES will converge to that new crit-
ical point with ψc>0, corresponding to the Hothouse
Earth state described in Ref. [3].
The actual evolution of the temperature (black os-
cillating line) is shown in Figure 6. The gray curve
corresponds to the associated equilibrium state which
evolves as hψi ∼ H1
3[1]. Hence, if the effect of the
human activities led to an increase of 1 K since the be-
ginning of the Anthropocene [7], about 50 years ago,
then 100 years after the start of the Anthropocene we
can expect a temperature increase to 3
√2=1.26 K.
This is, of course, assuming that the growth of H
remains linear.
3. Hothouse Earth state and other minima
We have seen in the previous section that a critical
point of the dynamical system corresponds necessarily
to an ES trajectory towards a minimum where the
ψ
p
Figure 5: Trajectory of the ES in the phase space for H(t) = t
and initial conditions (ψ, p)(0) = (0,0) (black line), modelling
the departure from the Holocene equilibrium, with the inclusion
of a Rayleigh dissipative term in the Lagrangian. Gray lines
show evolution for different initial conditions to show how they
all converge to the same moving equllibrium point.
t
ψ
Figure 6: Evolution of the ES temperature, ψ, and a function
of time for H(t) = tand initial conditions (ψ, p)(0) = (0,0)
(black line), modelling the departure from the Holocene equi-
librium, with the inclusion of a Rayleigh dissipative term in
the Lagrangian. The gray line represents the evolution of the
equilibrium point.
5

temperature is greater that the one at the Holocene
equilibrium.
As discussed in Ref. [3], this increase in the global
temperature can lead to a chain failure of the main
regulatory ecosystems of the ES that already show
tipping point features. It is therefore quite relevant to
investigate the possibility of engendering alternative
trajectories to the ES. This can be achieved through
the engineering of metastable minima on the phase
space, that is, new terms proportional to ψ2and/or
ψ3in the free energy function, Eq. (1) that force the
ES to remain close to the Holocene minimum.
Another strategy would be to consider the change
in the sign of the human drivers, H. In the context of
the planetary boundaries framework [4], the state of
the ES is specified through a set of 9 parameters and
the Holocene-like conditions are ensured provided the
ES remains within the so-called Safe Operating Space
(SOS) [5].
Indicating the effect of the human activity in alter-
ing the optimum Holocene conditions by hi, the bulk
of the human intervention, H, can be written as
H=
9
X
i=1
hi+
9
X
i,j=1
gij hihj+
9
X
i,j,k=1
αijk hihjhk+. . . ,
(14)
where the second and third set of terms indicate the
interaction between the various effects of the human
action on the planetary boundary parameters. Of
course, higher order interactions terms can be con-
sidered, but we shall restrict our considerations up
to second order and, in fact, to a subset of plane-
tary boundary parameters. It is physically reason-
able and mathematically convenient to assume that
the 9×9matrix, [gij ]is symmetric, gij =gji , and
non-degenerate, det[gij ]6= 0.
Let us consider only a couple of parameters, say
h1and h9, and assume, in particular, that the ninth
parameter corresponds to the Technosphere, i.e., the
set of human technological activities aiming to repair
or to mitigate the action on the variables away from
the SOS. Thus, under these conditions, we can write
Has
H=h1+h9+ 2g19h1h9+g11 h2
1+g99h2
9.(15)
It is easy to see that if h1>0,h9<0and g19 >0,
then the effect is to mitigate the destabilizing effect of
h1. The net effect of this technological interaction is
to ensure that the minimum due to human interven-
tion is, as discussed in Ref. [1], closer to the Holocene,
Figure 7: Stability landscape of the ES in terms of ψand H
with a metastable state.
minimising hψi, given that hψiis proportional to the
cubic root of H. We could also argue that g11 and g99
are negative too, given they can have an inhibiting
effect on affecting themselves.
The above considerations assume that the hiterms
do not depend on the temperature field. However,
it is most likely, in physical terms, that hi=hi(ψ),
meaning in fact that the effect of the human activities
might alter the free energy introducing new quadratic
and cubic terms. The effect of these terms would de-
pend, of course, on their relative strength, but may
lead to the appearance of a metastable state, as illus-
trated on the left side of the plot in Figure 7 for a ψ3
type term.
4. Conclusions
Starting from the physical framework provided by
the Landau-Ginzburg theory, we have built a descrip-
tion of the ES transition from the Holocene to the
Anthropocene as a dynamical system. That way, we
have a mathematical model well grounded on physics
that can capture the behaviour of the ES during this
transition, including its non-linearities and more com-
plex properties.
We thus use the Hamiltonian formulation, ubiqui-
tous in most branches of physics, to obtain the evo-
lution equations of the ES and examine its orbits in
the phase space, the space of all possible states of
the system. It then becomes evident how the increase
in human activities progressively deforms the phase
space of the ES towards higher temperatures.
With this kind of mathematical description of the
ES we can start to look for some of its proper-
ties, namely the existence and stability of equilibrium
points. Indeed, we show how a given evolution of the
6

human influence will affect these points. Even assum-
ing that their effects are bounded, the ES will unques-
tionably progress towards an equilibrium away from
the Holocene, supporting a Hothouse Earth scenario.
A key issue remains in how to correctly describe
the human effects in their multiple components. We
stress how these components are not independent and
may have direct (second-order) or higher order inter-
actions. We show, at least theoretically, that interac-
tions among these components, most particularly the
one associated with the Technosphere, can allow for
mitigating strategies in what concerns the inevitable
evolution towards a Hothouse Earth, as well as the
“engineering” of metastable states where the ES can
remain temporarily in equilibrium in a cooler state.
Acknowledgements
The authors would like to thank Will Steffen for
the insightful discussions.
References
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